面试数学题录

1. 数论与整数 (Numbers) (100题)

Prove that $n^2(n^2 - 1)(n^2 - 4)$ is divisible by 360 whenever $n$ is a natural number.

Prove that $n^2 - 1$ is divisible by 8 when $n$ is odd.

Prove that $n^5 - n$ is divisible by 6 whenever $n$ is a natural number.

Prove that $n^5 - n$ is divisible by 30 whenever $n$ is a natural number.

Prove that $4^n - 1$ is divisible by 3 whenever $n$ is a natural number.

Prove that $14^n + 11$ is never prime.

Let $n$ be a natural number. Suppose $a^n - 1$ is prime. Show that $a = 2$ and that $n$ must be prime (Mersenne Primes). Comment on primes of the form $2^n + 1$ (Fermat Numbers).

Find all prime numbers $p$ such that $2p - 1$ and $2p + 1$ are also prime.

Is $1234567891011$ a square number? Is $24681012141618202224$?

10.

Prove, without directly calculating its value, that $11^{10} - 1$ is divisible by 100.

11.

Prove that $1^{99} + 2^{99} + 3^{99} + 4^{99} + 5^{99}$ is divisible by 5.

12.

Prove $n^7 - n$ is always divisible by 42.

13.

What is the last digit of $3^{50}$?

14.

Is $x^2 + 2$ ever divisible by 5?

15.

Prove $4^n - 1$ is a multiple of 3.

16.

If $x^2 + y^2 = z^2$, prove $xyz$ is a multiple of 60.

17.

Can $1000003$ be written as the sum of 2 square numbers?

18.

Show that when you square an odd number, you always get one more than a multiple of 8.

19.

What are the possible unit digits for perfect squares?

20.

What are the possible remainders when a cube is divided by 9?

21.

Prove that $801,279,386,104$ can't be written as the sum of 3 cubes.

22.

What are the last two digits of the number which is formed by multiplying all the odd numbers from 1 to $1000000$?

23.

Prove that $1! + 2! + 3! + \ldots + n!$ has no square values for $n > 3$.

24.

Prove for $a^2 + b^2 = c^2$, where $a, b, c$ are positive integers, then $a$ and $b$ can’t both be odd.

25.

Show that $n^5 - n^3$ is divisible by 12.

26.

Suggest prime factors of $612612503503$.

27.

Prove that the product of four consecutive integers is divisible by 24.

28.

Prove that $n^2 \equiv (n+7)^2 \pmod{7}$.

29.

Find a positive integer $a$ such that $n^4 + a$ isn't prime for all integers $n$.

30.

Let $n > 6$ be an integer such that $n-1$ and $n+1$ are prime. Prove that $720 \mid n^2(n^2 + 16)$.

31.

Show that no number in the sequence $11, 111, 1111, 11111, \ldots$ is a perfect square.

32.

Let $x = 2^\alpha$, by rearranging the digits of $x$, can we get a different number $y = 2^k$?

33.

Prove that $n+3$ and $n^2 + 3$ cannot be both squares.

34.

Find the last two digits of $99^n$.

35.

Let $a$ be the integer consisting of $m$ digit 1's and $b$ be the integer consisting of a digit 1 at the start, a digit 5 at the end and with $(m-1)$ digit 0's in between.

36.

Show that $(ab + 1)$ is a perfect square and find its square root.

37.

Prove that $10201$ is composite in any base.

38.

Two positive numbers, $a$ and $b$, with distinct first digits are multiplied together. Is it possible for the first digit of the product to fall strictly between the first digits of the two numbers?

39.

What is the last digit of $36!$? How many zeros are on the end of $36!$?

40.

How many zeros does $371!$ have on the end of it?

41.

Write down 3 digits, and then write the number again next to itself, e.g.\ $145145$. Why is it divisible by $13$?

42.

How many $0$'s are in $100!$?

43.

How many zeros in $365!$?

44.

Is $0.\overline{9} = 1$? Why? Prove it.

45.

If $n$ is a perfect square and its second last digit is $7$, what are the possibilities for the last digit of $n$ and can you show this will always be the case?

46.

What is the greatest value of $n$ for which $20!$ is divisible by $2^n$?

47.

How many digits has $2^k$?

48.

Are there any integer solutions to the equation $x^2 + y^2 = 3z^2$ where $x, y, z$ are co-prime? Are there any integer solutions at all?

49.

Sketch $x^2 - ny^2 = 0$ where $n$ is a natural number.

50.

Find all natural solution pairs $(x, y)$ in the case $n = 9$.

51.

Find all natural solution pairs $(x, y)$ in the case $n = 10$.

52.

How many natural number solutions are there to the equation $\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = 1$ where $a < b < c$?

53.

Find all positive integer solutions $(x, y)$ to $x^2 + y^2 = 2015$.

54.

Will the following equation have any positive integer solutions $x^2 + 33y^2 = 555555555$?

55.

If $n, x, y, z$ are all positive integers, find all solutions of the equation $n^x + n^y = n^z$.

56.

Prove that the only solution to the equation $x^2 + y^2 + z^2 = 2xyz$ for integers $x, y, z$ is $x = y = z = 0$.

57.

If $x$ and $y$ are positive integers, find all solutions of the equation $2xy - 4x^2 + 12x - 5y = 11$.

58.

A right angled triangle has all of its sides an integer length. If the length of the perimeter equals the area, find all such triangles.

59.

Show that, if $n$ is an integer such that $(n-3)^3 + n^3 = (n+3)^3$, then $n$ is even and $n^2$ is a factor of $54$. Deduce that there is no integer $n$ which satisfies the equation (*).

60.

Is $\log_2(3)$ rational? Prove it.

61.

Is $\tan(1^\circ)$ irrational? What about $\cos(1^\circ)$?

62.

Prove that $\sqrt{3}$ is irrational.

63.

Can you show that any $\sqrt{p}$ is irrational for $p$ prime?

64.

Given $n$ consecutive positive integers, show that $n!$ is a factor of their product.

65.

Find the number of integer solutions to the equation $|x| + |y| \leq 100$.

66.

Are there any integer solutions to the equation $x^2 + y^2 = 3z^2$ where $x, y, z$ are co-prime?

67.

Are there any integer solutions at all?

68.

If three positive real numbers $a, b, c$ satisfy the following equations, show that at least one of them must

69.

be 1 and hence deduce all solutions: \[ abc = 1 \quad \text{and} \quad a + b + c = 3 = \frac{1}{a} + \frac{1}{b} + \frac{1}{c} \]

70.

Prove that the only solution to the equation $ x^2 + y^2 + z^2 = 2xyz $ for integers $ x, y, z $ is $ x = y = z = 0 $.

71.

If $ x $ and $ y $ are positive integers, find all solutions of the equation \[ 2xy - 4x^2 + 12x - 5y = 11. \]

72.

Prove that $ n^2(n^2 - 1)(n^2 - 4) $ is divisible by 360 whenever $ n $ is a natural number.

73.

Do you know a solution to the equation $ 5^x = 4^x + 3^x $? Are there any more? Prove it.

74.

Prove that $ n^2 - 1 $ is divisible by 8 when $ n $ is odd.

75.

Prove that $ n^5 - n $ is divisible by 6 whenever $ n $ is a natural number.

76.

Prove that $ n^5 - n $ is divisible by 30 whenever $ n $ is a natural number.

77.

Prove that $ 4^n - 1 $ is divisible by 3 whenever $ n $ is a natural number.

78.

Prove that there are infinitely many primes.

79.

Prove that there are infinitely many primes of the form $ 4n + 3 $.

80.

Is $ \log_2 3 $ rational? Prove it.

81.

Prove that $ 14^n + 11 $ is never prime.

82.

Let $ n $ be a natural number. Suppose $ a^n - 1 $ is prime. Show that $ a = 2 $ and that $ n $ must be prime (Mersenne Primes). Comment on primes of the form $ 2^{n+1} $ (Fermat Numbers).

83.

Find all prime numbers $ p $ such that $ 2p - 1 $ and $ 2p + 1 $ are also prime.

84.

Show that $ 3 < \pi < 4 $.

85.

If a natural number $ n $ has $ N $ digits, how many digits can $ n^2 $ have? What about $ n^n $?

86.

How would you write a formula for the number of digits of $ n $?

87.

Is $\tan(1^\circ)$ irrational?

88.

What about $\cos(1^\circ)$?

89.

Prove, without directly calculating its value, that $11^{10} - 1$ is divisible by $100$.

90.

Prove that $1^{99} + 2^{99} + 3^{99} + 4^{99} + 5^{99}$ is divisible by $5$.

91.

Let $a$ be the integer consisting of $m$ digit $1$'s and $b$ be the integer consisting of a digit $1$ at the start, a digit $5$ at the end and with $m-1$ digit $0$'s in between. Show that $ab + 1$ is a perfect square and find its square root.

92.

How many integers from $1$ to $10^{30}$ inclusive are perfect squares, cubes or fifth powers?

93.

$a_i \in \{-1, 1\}$

94.

$a_1a_2 + a_2a_3 + \cdots + a_n a_1 = 0$

95.

Prove $4 \mid n$.

96.

Prove if $2^n - 1$ is a prime, then $n$ must be a prime.

97.

If $2^n + 1$ is a prime, find a condition for $n$. If $n$ is odd, $2^n + 1 = (2 + 1)[\cdots]$ not prime except for $n = 1$. $n$ has no odd factor, $n = 2^k$, $k \geq 0$.

98.

Prove $n^2(n^2 - 1)(n^2 - 4)$ is divisible by $360$.

99.

Prove $14^n + 11$ is never a prime.

100.

For $s \in \mathbb{R}$, define $f(s) = \sum_{n=1}^\infty \frac{1}{n^s}$, $s$ is fixed, consider a positive integer valued random variable $X$. $P(x = n) = \frac{1}{n^s} \cdot \frac{1}{f(s)}$, $n \geq 1$.

2. 概率与计数 (Prob & Count) (100题)

If a round table has $n$ people sitting around it, what is the probability of person $A$ sitting exactly $k$ seats away from person $B$?

How can you maximise the number of regions $n$ straight lines will divide the plane into?

What is this maximum in terms of $n$?

What if we replace lines with circles?

What does this tell you about Venn Diagrams?

How many vertices and edges does a line segment have? A square? A cube? A tesseract?

Can you conjecture formulae for the number of edges and vertices of an $n$ dimensional hypercube?

Can you give the coordinates of the vertices of a tesseract (where 4 edges coincide with the coordinate axes)?

What would the longest length between two vertices be?

10.

In a football competition where every round played is a knockout match (i.e. a draw leading to a replay is not an option), how many matches will be played in the competition in total if there are $n$ teams?

11.

In how many ways can $2n$ opponents be paired in the first round of a tennis competition?

12.

Can you come up with a more succinct expression for your previous solution (i.e. if I gave you a large value for $n$ you could then use your calculator to calculate the answer quickly)?

13.

How many distinct tessellations of the plane use only one regular polygon?

14.

Why are there only five platonic solids?

15.

Using Euler's Polyhedron Formula $V - E + F = 2$ show that a platonic solid made of triangles must have 4, 8 or 20 faces.

16.

If I colour three faces of a cube red and the other faces blue, how many distinguishable colourings are there?

17.

Two full decks of cards are shuffled and placed side by side. I take the top card from each pile and pair them up. What is the probability I have:

i)
52 matching pairs?
51 matching pairs?
50 matching pairs?
49 matching pairs?
$k$ matching pairs?

18.

A thin rod is broken into three pieces. What is the probability that a triangle can be formed from the three pieces?

19.

A natural number from 1 to 1000000 is selected at random, what is the probability its cube ends in 11?

20.

If you pick 3 cards from a randomly shuffled pack of cards, are you more likely to see a face card than not?

21.

Alice and Bob play a fair game repeatedly for $\ell^1$ a game. If originally Alice has $\ell^a$ and Bob has $\ell^b$, what is Alice’s chance of winning all of Bob’s money, assuming that play continues until one person has lost all of his or her money?

22.

There are $n!$ permutations $(s_1, s_2, s_3, \ldots, s_n)$ of $(1,2,3,\ldots,n)$. How many of them satisfy $s_k > k - 3$ for $k = 1,2,3,\ldots,n$?

23.

A bracelet is made up of a combination of 11 red, yellow or blue beads. How many distinct bracelets can be made if you have at least 11 beads of each colour and if rotations are considered the same but reflections are not?

24.

i)
A regular fair dice is rolled twelve times, what is the probability of getting two of each number?
A fair ten-sided dice is rolled four times, what is the probability that your sequence of rolls is increasing?

25.

Toss a fair coin and stop when 4 consecutive $H$ appears. Find $E$ (number of toss).

26.

How many 1 are there in $1 \sim 1,000,000$?

27.

You are standing near the Pentagon. Given you are standing far enough away that you are not only seeing one side, what is the probability of you seeing only two sides? What if you could also be close enough to see just one side?

28.

If we have 25 people, what is the likelihood that at least one of them is born each month of the year?

29.

There are $2^n$ players, in first round each two meet and the winner enters the second round, two players are chosen randomly before the game starts. Find the probability of

$P$ (two of them meet in the first round)
$P$ (meet in the last round)
$P$ (meet in any round)

30.

There are $n$ red balls and $m$ blue balls in a bin.

(i)
If the person takes one ball from the bin, what is the probability that the ball is red? Take two balls, what is the probability both are red? One is red?
with replacement.
without replacement.

31.

For an $8 \times 8$ grid, if two adjacent boxes (top, bottom, left and right only) are coloured, the box will also be coloured. What is the smallest number of boxes coloured at the start to ensure the whole grid is coloured eventually?

32.

33.

34.

Start at point 2, each time move 1 unit. Let $P(n)$ denote the number of paths of length $n$. What can you find about $P(n)$? What does this tell you about the value of $P(n)$?

35.

There are $n$ points on the circumference of the circle. Find the probability that they are on the same semicircle.

36.

Choosing 3 points randomly in a unit square. Find the probability that the triangle formed has its side intersecting the diagonal of the square.

37.

Choosing 3 points randomly in a unit square. Find the probability that the side of triangle intersecting either $AB$ or $AC$ or both.

38.

39.

40.

Two players are tossing coins. Player 1 tosses $(n+1)$ coins and player 2 tosses $n$ coins. Find the probability that the number of $H$ that player 1 got is more than player 2.

41.

There are 24 green balls and 36 red balls; take out one each time and stop when there is no green ball. Find the probability that the box is empty.

42.

There is a pile of 129 coins on a table. 128 of them are unbiased and the remaining coin has heads on both sides. David chooses a coin at random and tosses it eight times. The coin comes up heads every time. What is the probability that it will come up heads the ninth time as well?

43.

Twenty balls are placed in an urn. Five are red, five green, five yellow and five blue. Three balls are drawn from the urn at random without replacement. What are the probabilities of the following events:

44.

(a) Exactly one of the balls drawn is red; (b) The three balls drawn have different colours; (c) The number of blue balls drawn is strictly greater than the number of yellow balls drawn.

45.

In the town of Rejectbridge, $\frac{2}{3}$ of the adult men are married to $\frac{3}{5}$ of the adult women. Given that each man is married to one woman, what fraction of the adult population of Rejectbridge is married?

46.

Suppose an unbiased six-sided die is rolled 6 times. What is the probability that the rolls are strictly increasing?

47.

An unbiased six-sided die is rolled $n$ times. What is the probability that the die has landed on each face at least once?

48.

(a) Find the number of ways in which a total of 10 may be obtained by throwing 3 unbiased six-sided dice. (b) In a game 4 players roll in turn an unbiased six-sided die until six is obtained. The winner is the first to roll a six. Find the probability that (i) the first player wins. (ii) the last player wins.

49.

Oliver’s stamp collection consists of three books. Two tenths of his stamps are in the first book, several sevenths in the second book and there are 303 stamps in the third book. How many stamps does Oliver have in total?

50.

There are $b$ black chess pieces and $w$ white chess pieces in a bag. A person takes out one chess piece at a time. What is the probability that the last chess piece is white?

51.

Two players, A and B, play a game where a biased coin is flipped with a probability $p$ of coming up heads. A player wins if they get heads. If player A gets tails then it is player B’s turn to flip the coin and if player B gets tails then it is again player A’s turn. Player A starts the game. Find the probability that player A wins.

52.

There are $n$ coins $c_1, c_2, c_3, \ldots, c_n$. Any coin $c_k$ is biased such that the probability of it coming up heads is $\frac{1}{2k+1}$. If all $n$ coins are tossed, what is the probability that the number of times heads has come up is odd?

53.

What is the expected number of times a six-sided die is rolled such that it lands on all faces at least once? What about an $n$-sided die?

54.

(Keble’s) There are 10 white balls and 1 black ball in an urn. Amy, sampling with replacement, failed

55.

to sample the black ball after 10000 rounds. She claims that this event is very unlikely to happen, with probability less than $0.0001$. Is she correct?

56.

(Balliol) Design a coin game that played by $A$ and $B$, the probability of $A$ wins is $\frac{1}{4}$.

57.

There is another coin game, the rule is: $A$ flips the coin first, $A$ loses when he gets tail. If $A$ gets head, $B$ flips the coin, $B$ loses when he gets tail as well. If $B$ gets head, it would be $A$'s turn again.

58.

What is the probability of $A$ wins?

59.

(St Hugh's) (a) How many regions can a circle be divided into by one line? two lines? three lines? (b) What is the maximum number of regions that a circle can be divided into by $n$ lines?

60.

If you have $n$ non-parallel lines in a plane, how many points of intersection are there?

61.

What is the most pieces of pizza I can get from $n$ cuts?

62.

There are 30 people in one room. What is the probability that of them have the same birthday?

63.

If a round table has $n$ people sitting around it, what is the probability of person $A$ sitting exactly $k$ seats away from person $B$?

64.

Problem 30: Two people are playing a game which involves taking it in turns to each chillies. There are 5 mild chillies and 1 hot chilli. Assuming the game is over when the hot chilli is eaten (and that I don't like hot chillies), is it a disadvantage to go first? What is the probability that I will eat the chilli if I go first? How about if there are 6 mild and 2 hot?

65.

There is a game with 2 players (A and B) who take turns to roll a die and have to roll a six to win. What is the probability of person A winning?

66.

Probability A winning $ = \frac{1}{6} + \left(\frac{5}{6}\right)^2 \frac{1}{6} + \left(\frac{5}{6}\right)^4 \frac{1}{6} + \ldots = \frac{1}{6} \sum_{k=0}^{\infty} \left(\frac{5}{6}\right)^{2k} = \frac{6}{11} $

67.

Probability B winning $ = \frac{5}{6} \left( \frac{1}{6} + \left(\frac{5}{6}\right)^2 \frac{1}{6} + \left(\frac{5}{6}\right)^4 \frac{1}{6} + \ldots \right) = \frac{5}{6} \left( \frac{1}{6} \sum_{k=0}^{\infty} \left(\frac{5}{6}\right)^{2k} \right) = \frac{5}{11} $

68.

Problem 62: What's the probability of flipping $n$ consecutive heads on a fair coin? What about an even number of consecutive heads?

69.

Probability of flipping $n$ consecutive heads $ = \left(\frac{1}{2}\right)^n $

70.

Probability of flipping even consecutive heads $ = \left(\frac{1}{2}\right)^2 + \left(\frac{1}{2}\right)^4 + \left(\frac{1}{2}\right)^6 + \ldots = \sum_{k=1}^{\infty} \left(\frac{1}{2}\right)^{2k} = \frac{1}{3} $

71.

Problem 64: A 10 digit number is made up of only 5’s and 0’s. It is also divisible by 9. How many possibilities are there for the number.

72.

Problem 88: If I have a chance $p$ of winning a point in tennis, what is the chance of winning a game. Assuming that the first player who wins four points wins the game.

73.

Then $P(\text{win in 4 points}) = p^4$

74.

$P(\text{win in 5 points}) = (4\text{ W and }1\text{ L, but the loss cannot be in the fifth point}) = \frac{4!}{3!1!} p^4 q$

75.

$P(\text{win in 6 points}) = (4\text{ W and }2\text{ L, but the loss cannot be in the sixth point}) = \frac{5!}{3!2!} p^4 q^2$

76.

$P(\text{win in 7 points}) = (4\text{ W and }3\text{ L, but the loss cannot be in the seventh point}) = \frac{6!}{3!3!} p^4 q^3$

77.

The winner of the game will be decided after playing 7 points.

78.

Probability of winning $ = p^4 + \frac{4!}{3!1!} p^4 q + \frac{5!}{3!2!} p^4 q^2 + \frac{6!}{3!3!} p^4 q^3 $

79.

There is a pile of 129 coins on a table, all unbiased except for one which has heads on both sides. Bob chooses a coin at random and tosses it eight times. The coin comes up heads every time. What is the

80.

probability that it will come up heads the ninth time as well?

81.

Twenty balls are placed in an urn. Five are red, five green, five yellow and five blue. Three balls are drawn from the urn at random without replacement. Write down expressions for the probabilities of the following events. (You need not calculate their numerical values.)

82.

(i) Exactly one of the balls drawn is red.

83.

(ii) The three balls drawn have different colours.

84.

(iii) The number of blue balls drawn is strictly greater than the number of yellow balls drawn.

85.

Six identical-looking coins are in a box, of which five are unbiased, while the sixth comes up heads with probability $3/4$ and tails with probability $1/4$. Three coins are chosen from the box at random and removed. One of those three is chosen at random and tossed three times, coming up heads every time. Given this information.

86.

(a) What is the probability that the final coin selected was the biased coin?

87.

(b) What is the probability that the biased coin is amongst the three coins removed from the box?

88.

A fair coin is tossed repeatedly. Let $t$ be the time at which we first see three consecutive heads (thus flips number $t-2$, $t-1$, $t$ are all heads), and let $s$ be the time at which we first see four consecutive heads. What is the probability that $s = t + 1$? And what is the probability that $s = t + 9$?

89.

1) What are the chances that with 6 people any of them celebrate their birthday in the same month? (Assume equal months)

90.

2) There are 30 people in a room ... what is the chance that any two of them celebrate their birthday on the same day? Assume 365 days in a year. (NRICH MATHEMATICS CAMBRIDGE)

91.

If we have 25 people, what is the likelihood that at least one of them is born each month of the year? (Oxford applicant, The Student Room) 25 个人当中要保证每个月至少生一个人的概率？

92.

Life or Death? The Emperor's Proposition. 国王的决策问题

93.

You are a prisoner sentenced to death. The Emperor offers you a chance to live by playing a simple game.

94.

He gives you 50 black marbles, 50 white marbles and 2 empty bowls. He then says, "Divide these 100 marbles into these 2 bowls. You can divide them any way you like as long as you use all the marbles. Then I will blindfold you and mix the bowls around. You then can choose one bowl and remove ONE marble. If the marble is WHITE you will live, but if the marble is BLACK, you will die." (NRICH MATHEMATICS CAMBRIDGE)

95.

If a round table has $ n $ people sitting around it, what is the probability of person $ A $ sitting exactly $ k $ seats away from person $ B $? (submitted by Oximple Applicant) 同学们注意这道题是 $ A $ 和 $ B $ 之间有非常精确的 $ k $ 个位置，实质上是一个对称问题.

96.

10 persons are seated at a round table. The number of ways of selecting 3 persons out of them if no two persons are adjacent to each other is? (www.quora.com)

97.

12 persons are seated around a round table. What is the probability that two particular persons sit together? (Submitted by Oxford Applicant, The Student Room)

98.

How many ways can 8 persons sit at a round table with 10 seats so that there is exactly one person between the two empty seats? (The Student Room)

99.

"Let's play a game of Russian roulette. You are tied to your chair. Here's a gun, a revolver. Here's the barrel of the gun, six chambers, all empty. Now watch me as I put two bullets into the barrel, into two adjacent chambers. I close the barrel and spin it. I put a gun to your head and pull the trigger. Click. Lucky you! Now I'm going to pull the trigger one more time. Which would you prefer: that I spin the barrel first or that I just pull the trigger?" (NRICH MATHEMATICS CAMBRIDGE) 这是一个非常有意思的俄罗斯转盘问题，用到条件概率的知识点

100.

If I have a chance $p$ of winning a point in tennis, what's the chance of winning a game? (Submitted by Oxford Applicant, The Student Room)

3. 组合数学 (Combinatorics) (100题)

How many ways can you arrange each of the digits 1 to 6 to create distinct 6 digit numbers?

How many of these contain the digits 1,2 and 3 next to each other and in that order?

In how many arrangements does 5 occur before 1 ?

How many distinct 6 digit numbers are there in which all of the digits 1 to 5 appear?

The Power Set is formed from all subsets of a given set.

If a set contains $n$ elements what is the cardinality of its Power Set?

How many subsets contain a given element $x_1$ ?

If a round table has $n$ people sitting around it, what is the probability of person $A$ sitting exactly $k$ seats away from person $B$ ?

How can you maximise the number of regions $n$ straight lines will divide the plane into?

10.

What is this maximum in terms of $n$ ?

11.

What if we replace lines with circles?

12.

What does this tell you about Venn Diagrams?

13.

14.

In how many ways can $2n$ opponents be paired in the first round of a tennis competition?

15.

Can you come up with a more succinct expression for your previous solution (i.e. if I gave you a large value for $n$ you could then use your calculator to calculate the answer quickly)?

16.

If I colour three faces of a cube red and the other faces blue, how many distinguishable colourings are there?

17.

Every subset of the set $(1,2,3,\ldots,n)$ either contains the element 1 or it does not.

18.

By considering these two possibilities, show that \[ \binom{n-1}{r-1} + \binom{n-1}{r} = \binom{n}{r} \]

19.

Explain why \[ \binom{n-2}{r-2} + 2\binom{n-2}{r-1} + \binom{n-2}{r} = \binom{n}{r} \]

20.

Two full decks of cards are shuffled and placed side by side. I take the top card from each pile and pair them up. What is the probability I have:

21.

i) 52 matching pairs? ii) 51 matching pairs? iii) 50 matching pairs? iv) 49 matching pairs? v) $k$ matching pairs?

22.

A thin rod is broken into three pieces. What is the probability that a triangle can be formed from the three pieces?

23.

Given $n$ consecutive positive integers, show that $n!$ is a factor of their product.

24.

A natural number from 1 to 1000000 is selected at random, what is the probability its cube ends in 11?

25.

How many integers from 1 to $10^{30}$ inclusive are perfect squares, cubes or fifth powers?

26.

If you pick 3 cards from a randomly shuffled pack of cards, are you more likely to see a face card than not?

27.

There are $n!$ permutations $(s_1, s_2, s_3, \ldots, s_n)$ of $(1,2,3,\ldots,n)$. How many of them satisfy $s_k > k - 3$ for $k = 1,2,3,\ldots,n$?

28.

29.

i) A regular fair dice is rolled twelve times, what is the probability of getting two of each number? ii) A fair ten-sided dice is rolled four times, what is the probability that your sequence of rolls is increasing?

30.

How many possible subsets are there of the set: $\{1,2,3,4,5,6,7,8\}$?

31.

There are three children, at least one of them is a girl, what is the probability that they are all girls?

32.

$N$ people are in a room, what is the probability that any two of them share a birthday?

33.

How many subsets of $\{a,b,c\}$ are there? How many subsets of $\{a_1,a_2,a_3,\ldots,a_n\}$ are there?

34.

If I had a cube and six colours and painted each side a different colour, how many (different) ways could

35.

I paint the cube? What about if I had $n$ colours instead of 6?

36.

3 girls and 4 boys were standing in a circle. What is the probability that two girls are together but one is not with them?

37.

How many squares can be made from a grid of ten by ten dots (ignore diagonal squares)?

38.

How many ways there are of getting from one vertex of a cube to the opposite vertex without going over the same edge twice?

39.

What's the probability of flipping $n$ consecutive heads on a fair coin? What about an even number of consecutive heads?

40.

A 10 digit number is made up of only 5s and 0s. It's also divisible by 9. How many possibilities are there for the number?

41.

Explain the 'Monty Hall' problem. [3 doors, 2 goats, 1 car prize.]

42.

Using three colours, in how many different ways can you colour a disc split into three equal portions?

43.

Two players $A$ and $B$ roll a die. The first to roll a six wins. Find $P(A \text{ wins})$. What about 3 players $A$, $B$ and $C$ with $P(A \text{ wins})$?

44.

How many multiples of 2013 have 2013 factors?

45.

If you have $n$ non-parallel lines in a plane, how many points of intersection are there?

46.

How many numbers satisfy conditions:

They have 3 digits.
All digits are even.
Digits are distinct.

47.

A thin hoop of diameter $d$ is thrown on to an infinitely large chessboard with squares of side $L$. What is the chance of the hoop enclosing two colours?

48.

An infinitely large floor is tiled with regular hexagonal tiles of side $L$. Different colours of tiles are used so that no two tiles of the same colour touch. A hoop of diameter $d$ is thrown onto the tiles. What is the

49.

chance of the hoop enclosing more than one colour?

50.

If we have 25 people, what is the likelihood that at least one of them is born each month of the year?

51.

The set $1, 2, \ldots, n$, can be written in the form of the concatenation of two sets in how many ways?

52.

There are 10 white balls and 1 black ball in an urn. Amy, sampling with replacement, failed to sample the black ball after 10000 rounds. She claims that this event is very unlikely to happen, with probability less than $0.0001$. Is she correct?

53.

Design a coin game that is played by $A$ and $B$, the probability of $A$ winning is $\frac{1}{4}$. There is another coin game, the rule is: $A$ flips the coin first, $A$ loses when he gets tail. If $A$ gets head, $B$ flips the coin, $B$ loses when he gets tail as well. If $B$ gets head, it would be $A$'s turn again. What is the probability of $A$ winning?

54.

There are four numbers in the range of 1 to 10 that are not a multiple of 3, 5, 7. How many numbers are not a multiple of 3, 5, 7 in the range of 1 to 1000?

55.

Lexicographic G, I, K, P, U. (A lexicographic order is an arrangement of characters, words, or numbers in alphabetical order.) List from GIKPU to UPKIG, where KIPUG?

56.

Define a new term balanced number, when one digit of a number is the average value of the other digits, this number is called a balanced number.

57.

(a) Write down three balanced numbers.

58.

(b) Write down the smallest nine balanced numbers between 100 and 999.

59.

60.

(d) How many balanced numbers have nine digits?

61.

(e) How many balanced numbers have $n$ digits?

62.

There are 6 ropes in a bag. In each step, two rope ends are picked at random, tied together, and put back into the bag. The process is repeated until there are no free ends.

63.

What is the expected number of loops $e_n$ at the end of the process? (Hint: Find a formula linking $e_n$ and $e_{n-1}$)

64.

A gambler played a game with his friend; he bet half of his money on the toss of a coin; he won on heads and lost on tails. The game was repeated over and over, and at the end the gambler had lost as many times as he had won. Did he make money, lose money, or break even?

65.

Alice and Bob play a fair game repeatedly for $1$ a game. If originally Alice has $a$ and Bob has $b$, what is Alice’s chance of winning all of Bob’s money, assuming that play continues until one person has lost all of his or her money?

66.

If I bet $5$ on you getting a heads when you flip a coin, what is a fair bet?

67.

If I bet $100$ on rolling a six, with a six sided die, what is a fair bet?

68.

You arrive at Rochester with $x$ in your bank account. Each time you visit the bank you withdraw half of what’s left, plus $1$ to donate to the appeal to erect a statue of Brian in front of the cathedral. How much is left after $i,\,1\quad ii,\,2\quad iii,\,3$ and $iv,\,n$ visits to the bank?

69.

What is the expected number of rolls of a $6$ sided die for all faces to appear at least once? What about an $n$ sided die?

70.

You play a game with a biased coin. If you get a head, you get a pound and the game continues; if you get a tails, the game ends (you keep your money from any past heads). What is the expected value? Can you do it with conditional expectation?

71.

The numbers $1$ to $1000$ are written on a blackboard. You randomly choose two numbers $a$ and $b$ from among them and replace them with their difference. You continue this process until you are left with a single number on the board, is it possible for you to be left with the number $1$?

72.

$10$ distinct points lie within a unit square, prove at least two of the points lie within $\frac{\sqrt{2}}{3}$ units of each other.

73.

Consider an infinite chessboard, the squares of which have been filled with positive integers. Each of these integers is the arithmetic mean of four of its neighbours (above, below, left, right). Show that all the integers are equal to each other.

74.

If $n$ points are distributed around the circumference of a circle and each point is joined to every other point by a chord of the circle (assuming that no three chords intersect at a point inside the circle), into how many regions is the circle divided?

75.

$2n$ points are chosen in the plane such that no $3$ are collinear, $n$ are coloured blue and $n$ are coloured red. Prove that it is always possible to join the $n$ red points to the $n$ blue points by line segments, such that no two line segments cross.

76.

For $ n > 1 $, the integers from $ 1 $ to $ n^2 $ are placed in the cells of an $ n \times n $ chessboard. Show that there is a pair of horizontally, vertically, or diagonally adjacent cells whose value differs by at least $ n+1 $.

77.

Say you have finitely many red and blue points on a plane with the interesting property: every line segment that joins two points of the same colour contains a point of the other colour. Prove that all the points lie on a single straight line.

78.

$ n $ students are standing in a field such that the distance between each pair is distinct. Each student is holding a ball, and when the teacher blows a whistle, each student throws their ball to the nearest student. Prove that there is a pair of students that throw their balls to each other.

79.

Every subset of the set $ \{1,2,3,\ldots,n\} $ either contains the element 1 or it does not. By considering these two possibilities, show that \[ \binom{n-1}{r-1} + \binom{n-1}{r} = \binom{n}{r}. \] Explain why \[ \binom{n-2}{r-2} + 2\binom{n-2}{r-1} + \binom{n-2}{r} = \binom{n}{r}. \]

80.

The T-tetromino is the shape made by joining four $ 1 \times 1 $ squares edge to edge, as shown. A rectangle $ R $ has dimensions $ (2a) \times (2b) $ where $ a $ and $ b $ are integers. The expression '$ R $ can be tiled by $ T $' means that $ R $ can be covered exactly by copies of $ T $ without gaps or overlaps.

81.

i) Can $ R $ be tiled by $ T $ when both $ a $ and $ b $ are even?

82.

ii) Can $ R $ be tiled by $ T $ when both $ a $ and $ b $ are odd?

83.

84.

85.

I have a large supply of counters which I place in each of the $ 1 \times 1 $ squares of an $ 8 \times 8 $ chessboard (1 counter on each square). Each counter is red, white or blue. A particular pattern of coloured counters is called an arrangement. Determine whether there are more arrangements which contain an even number of red counters or more arrangements which contain an odd number of red counters.

86.

Prove that it is impossible to have a cuboid for which the volume, the surface area and the perimeter are numerically equal (the perimeter of a cuboid is the sum of the lengths of all its twelve edges).

87.

A two player game is played on a $ 5 \times 5 $ grid. A token starts in the bottom left corner of the grid. On each

88.

turn, a player can move the token one or two units to the right, or to the leftmost square of the above row. The last player who is able to move wins. Determine which positions of the token are winning positions and which are losing. Generalize this problem to larger grids. How many winning positions are there on an $m \times n$ grid?

89.

Two people play a game: There are $n$ sweets in a pile and they each take it in turns to remove at least one sweet from the pile whilst ensuring they take no more than half of what remains. The person who removes the last sweet is the loser. Are there values of $n$ for which the second player has a winning strategy?

90.

Show that the family of concentric circles which have centre $\left(\frac{1}{3}, \sqrt{2}\right)$ are such that each circle has exactly 1 lattice point on its boundary, and each lattice point is on a circle.

91.

There are 6 ropes in a bag. In each step, two rope ends are picked at random, tied together and put back into a bag. The process is repeated until there are no free ends. What is the expected number of loops $e_n$ at the end of the process? (Hint: Find a formula linking $e_n$ and $e_{n-1}$)

92.

A three-dimensional version of noughts and crosses can be played with a $4 \times 4$ cube, the winner is the first player to get four noughts (or crosses) in a straight line. How many winning lines are there?

93.

A gambler played a game with his friend, he bet half of his money on the toss of coin; he won on heads and lost on tails. The game was repeated over and over and at the end the gambler had lost as many times as he had won. Did he make money, lose money or break even?

94.

How many ways can you arrange each of the digits 1 to 6 to create distinct 6-digit numbers?

95.

How many of these contain the digit 1, 2, and 3 next to each other and in that order?

96.

How many distinct 6-digit numbers are there in which all of the digits 1 to 5 appear?

97.

Prove for every four diagonal numbers their sum = 34 e.g. $(1,6,11,16)$ or $(2,7,12,13)$ \[

98.

If $n$ points are distributed around the circumference of a circle and each point is joined to every other

99.

point by a chord of the circle (assuming that no three chords intersect at a point inside the circle) into how many regions is the circle divided?

100.

There are $ n $ ropes in a bag. In each step, two rope ends are picked at random, tied together and put back into a bag. The process is repeated until there are no free ends. Find a formula linking $ e_n $ and $ e_{n-1} $.

4. 离散数学 (Discrete) (100题)

The numbers 1 to 1000 are written on a blackboard. You randomly choose two numbers $a$ and $b$ from among them and replace them with their difference.

You continue this process until you are left with a single number on the board, is it possible for you to be left with the number 1?

10 distinct points lie within a unit square, prove at least two of the points lie within $\frac{\sqrt{2}}{3}$ units of each other.

$2n$ points are chosen in the plane such that no 3 are collinear, $n$ are coloured blue and $n$ are coloured red. Prove that it is always possible to join the $n$ red points to the $n$ blue points by line segments, such that no two line segments cross.

For $n > 1$, the integers from 1 to $n^2$ are placed in the cells of an $n \times n$ chessboard. Show that there is a pair of horizontally, vertically, or diagonally adjacent cells whose value differs by at least $n+1$.

$n$ students are standing in a field such that the distance between each pair is distinct. Each student is holding a ball, and when the teacher blows a whistle, each student throws their ball to the nearest student. Prove that there is a pair of students that throw their balls to each other.

A longevity chain is a sequence of consecutive integers, whose digit sums are never a multiple of 9. What is the longest possible length of a longevity chain?

10.

The $T$-tetromino is the shape made by joining four $1 \times 1$ squares edge to edge, as shown.

11.

12.

13.

A rectangle $ R $ has dimensions $ (2a) \times (2b) $ where $ a $ and $ b $ are integers. The expression `R can be tiled by $ T $' means that $ R $ can be covered exactly by copies of $ T $ without gaps or overlaps.

14.

i) Can $ R $ be tiled by $ T $ when both $ a $ and $ b $ are even?

15.

ii) Can $ R $ be tiled by $ T $ when both $ a $ and $ b $ are odd?

16.

17.

A two player game is played on a $ 5 \times 5 $ grid. A token starts in the bottom left corner of the grid. On each turn, a player can move the token one or two units to the right, or to the leftmost square of the above row. The last player who is able to move wins. Determine which positions of the token are winning positions and which are losing.

18.

Generalize this problem to larger grids. How many winning positions are there on an $ m \times n $ grid?

19.

Two people play a game: There are $ n $ sweets in a pile and they each take it in turns to remove at least one sweet from the pile whilst ensuring they take no more than half of what remains. The person who removes the last sweet is the loser. Are there values of $ n $ for which the second player has a winning strategy?

20.

Show that the family of concentric circles which have centre $ \left( \frac{1}{3}, \sqrt{2} \right) $ are such that each circle has exactly 1 lattice point on its boundary, and each lattice point is on a circle.

21.

Construct a counter example to the statement: When written in decimal notation, every square number

22.

has at most 1000 digits that are not 0 or 1.

23.

Show that if $n$ is a positive integer greater than one then $\frac{1}{1} + \frac{1}{2} + \cdots + \frac{1}{n}$ is not an integer.

24.

25.

A teacher gives plain cards to a class and tells each student to draw a number on one side and a letter on the other. The rule is that if the number is even, then the letter must be a vowel. Which of these cards must be checked (by turning over) to see if the rule was followed?

26.

Find a series of consecutive integers such that the sum of the series is a power of 2.

27.

Two people are playing a game which involves taking it in turns to eat chillies. There are 5 mild chillies and 1 hot chilli. Assuming the game is over when the hot chilli is eaten (and that I don’t like hot chillies), is it a disadvantage to go first? What is the probability that I will eat the chilli if I go first? How about if there are 6 mild and 2 hot?

28.

50 people go to a party and shake hands with a random number of people, no one shakes the same person’s hand twice. Is it possible no two people shake hands the same number of times?

29.

Suppose there’s an invisible rabbit which is moving along a train track from an unknown initial position and it jumps by an unknown amount every second. Every second, before it jumps, you can pick a point on the train track to try catch the rabbit. Prove that you can catch it in a finite time.

30.

Take 5 points in an equilateral triangle of side length 1. Prove that there are two of them at a distance of not greater than 0.5.

31.

Consider a square of side length 2. If we have 9 points inside it, do three of the points necessarily form a triangle of area less than $1/2$?

32.

Given 5 points on a sphere, can we split the sphere in two so that 4 points lie on one of the hemispheres?

33.

There are $ n $ ants on a rope of length 10 m. They all move at a speed of 10 m/s. If two ants bump into each other on the rope, they instantly reverse their direction and move at the same speed. The ants can be facing in any direction. If an ant gets to the end of the rope then it falls off, prove that they all fall off eventually. Can you give an upper bound on the amount of time until all the ants fall off the rope?

34.

There are 50 ants on a 10 m line. The 25 left-most ants are moving right, and the 25 right-most ants are moving left. When 2 ants collide, they will both reverse direction. How many collisions will there have been in total once all ants have fallen off the end of the line?

35.

You have $ n $ coins $ C_1, C_2, \ldots, C_n $. Each coin $ C_k $ is biased so that tossing a heads with coin $ C_k $ has a probability of $ \frac{1}{2^{k+1}} $. If all $ n $ coins are tossed, what is the probability that the number of heads is odd?

36.

Is there an infinite sequence of real numbers $ a_1, a_2, \ldots $ such that $ a_1^m + a_2^m + \cdots = m $ for every positive integer $ m $?

37.

There's a torus/ring doughnut shaped space station with 2 spacemen on a spacewalk standing diametrically opposite each other. Can then ask a variety of questions such as if spaceman A wants to throw a spanner to spaceman B, what angle and speed should they choose (stating any assumptions made, e.g. that gravity = 0)?

38.

A telephone company has run a very long telephone cable all the way round the middle of the earth. Assuming the Earth to be a sphere, and without recourse to pen and paper, estimate how much additional cable would be required to raise the telephone cable to the top of the 10 m tall telephone poles.

39.

Is it possible to cover a chess-board with dominoes, when two corner squares have been removed from the chessboard and they are (a) adjacent corners, or conversely, (b) diagonally opposite.

40.

Determine all pairs $ (m,n) $ of positive integers satisfying:

(a)
two of the digits of $ m $ are the same as the corresponding digits of $ n $, while the other digit of $ m $ is 1 less than the corresponding digit of $ n $ (as in, say, 263 and 273);
both $ m $ and $ n $ are three-digit squares.

41.

A typical hint from the tutor would be to consider this case for 3 digit pairs $ (m,n) $ that is, Find as efficiently as possible all pairs $ (m,n) $ of positive integers satisfying the following two conditions:

(a)
two of the digits of $ m $ are the same as the corresponding digits of $ n $, while the other two digits of $ m $ are both 1 less than the corresponding digits of $ n $;

42.

(b) both $m$ and $n$ are four-digit squares.

43.

Consider a $2$ by $2$ matrix \[ A = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix} \] What is $A^n$?

44.

Given a set of $10$ natural numbers from $1$ to $100$. Prove that there are two disjoint non-empty subsets having the same sum.

45.

For natural number we define $3$ rules:

At least one white and one red number.
red + white = white.
red $\times$ white = red.

Question: (a) Is $0$ red or white? (b) Is $1$ red or white?

46.

Define a kind of set selfish, write down a selfish and a non-selfish set. (when there is number in the set equals to the number of elements in the set, the set is a selfish)

47.

Find the number of subsets of set $(1,2,3,4,\ldots,n)$ which is minimal selfish. (A set is minimal selfish when it is selfish and has no non-selfish subsets.)

48.

How many ways can you arrange each of the digits $1$ to $6$ to create distinct $6$ digit numbers?

49.

How many of these contain the digits $1,2$ and $3$ next to each other and in that order?

50.

In how many arrangements does $5$ occur before $1$?

51.

How many distinct $6$ digit numbers are there in which all of the digits $1$ to $5$ appear?

52.

The Power Set is formed from all subsets of a given set.

53.

If a set contains $n$ elements what is the cardinality of its Power Set?

54.

How many subsets contain a given element $x_1$?

55.

A longevity chain is a sequence of consecutive integers, whose digit sums are never a multiple of $9$. What is the longest possible length of a longevity chain?

56.

For each non-empty subset of integers $(1,2,3,\ldots,n)$ consider the reciprocal of the product of the elements.

57.

Let $ S_n $ denote the sum of these products. Conjecture and prove a formula for $ S_n $.

58.

A right angled triangle has all of its sides an integer length. If the length of the perimeter equals the area, find all such triangles.

59.

Find the last two digits of $ 99^n $.

60.

Which will be larger as $ n \to \infty $: $ 2^{2^{2^n}} $ or $ 100^{100^n} $?

61.

Consider the sequence $ 0, 1, 1, 2, 2, \ldots, r, r, r+1, r+1, \ldots $, deduce the sum of the first $ n $ terms $ S(n) $. Prove that $ S(s + t) - S(s - t) = st $ where $ s $ and $ t $ are positive integers and $ s > t $.

62.

Prove that for $ n $ a positive integer, \[ 1 + \frac{1}{1!} + \frac{1}{2!} + \cdots + \frac{1}{n!} < 3. \]

63.

Prove that for any positive integer $ n $ and any real number $ x $, \[ \left\lfloor \frac{\lfloor nx \rfloor}{n} \right\rfloor = \lfloor x \rfloor, \] where $ \lfloor z \rfloor $ denotes the largest integer value less than or equal to $ z $.

64.

Two positive numbers, $ a $ and $ b $, with distinct first digits are multiplied together. Is it possible for the first digit of the product to fall strictly between the first digits of the two numbers?

65.

$ F_1 = 1 $, $ F_2 = 1 $, $ F_n = F_{n-1} + F_{n-2} $, find the parity of $ F_{100} $.

66.

Given that $ n $ is a natural number, compare $ n^{n+1} $ and $ (n+1)^n $.

67.

$ P(D_k) = \frac{1}{k^2} $. Show the events $ \{D_p : p \text{ prime}\} $ are independent. For random variables $ A $ and $ B $, they are independent if and only if \[ P(A) \cdot P(B) = P(A \cap B). \] For random variables $ A_1, A_2, \ldots, A_n $, they are independent if and only if \[ P\left( \bigcap_i A_i \right) = \prod_i P(A_i). \] If $ \{A_i : i \in I\} $ are independent, then $ \{A_i^c : i \in I\} $ are independent.

68.

69.

How would you write a formula for the number of digits of $ n $?

70.

Find the sum of the coefficients of \[ (1 - 3x + 3x^2 - 5x^3 + 5x^4)(1 + 3x - 3x^2 + 5x^3 - 5x^4). \]

71.

A number is a power of $ 2 $. Prove that by permuting its digits, we cannot obtain another power of $ 2 $.

72.

There are 10 two-digit numbers. Prove that there exist two disjoint subsets with the same sum.

73.

Find $ d\left[d\left[d\left(4444^{4494}\right)\right]\right] $, where $ d(x) $ is the sum of the digits of $ x $.

74.

How many integer points are there in a circle of radius $ r $?

75.

Let $ M = \{1, 2, \cdots, 24\} $, and let $ S $ be a subset with 10 distinct elements $ x_1, x_2, y_1, y_2 \in S $. Prove that there exist $ x_1 + x_2 = y_1 + y_2 $.

76.

Binary sequences allow 01, 10, 11, but not 00, and cannot start with 0. Let $ A_n $ denote the number of such sequences of length $ n $. Find a recurrence relation for $ A_n $.

77.

\[ A_n = A_{n-1} + A_{n-2} \]

78.

Is there a polynomial $ P(n) = \{P(n) : n \in \mathbb{Z}\} $ such that $ \forall n \geq N $, $ P(n) $ is prime?

79.

(1) Prove that for any polynomial $ f(x) $, $ x - y \mid f(x) - f(y) $.

80.

(2) If we assume that $ P(N) = q $ is prime, consider $ P(kq + N) $.

81.

(3) Why can't $ P(n) = q $ for all $ n \geq N $?

82.

Assume $ a $ is a fixed positive integer, and $ x $ is a positive integer. Suppose $ \left\lfloor \frac{x}{a} \right\rfloor = \left\lfloor \frac{x}{a-1} \right\rfloor $. How many values of $ x $ satisfy this condition?

83.

Prove that $ 100\sqrt{3 + \sqrt{2}} + 100\sqrt{3 - \sqrt{2}} $ is irrational.

84.

Find all sets $ A $ (where $ A \subseteq \mathbb{Z} $) satisfying the property: for all $ x, y \in A $, $ x - y \in A $ (*).

85.

(1) What are the finite sets $ A $?

86.

(2) Prove that $ \{0\} $ and $ \emptyset $ are the only finite sets satisfying (*).

87.

(3) Is there any infinite set $ A $ other than $ \mathbb{Z} $ satisfying (*)?

88.

(4) There exists some $ b \in \mathbb{Z} $ such that $ A_b = \{bk \mid k \in \mathbb{Z}\} $ satisfies (*). What is the relationship between $ a $ and $ b $, given that $ a \in A $?

89.

(5) Prove that there are no other sets $ A $ satisfying (*)?

90.

For a sequence $ A_0 \sim A_n $, for any $ i $, there are exactly $ A_i $ elements equal to $ i $. Find all possible sequences of length 2024.

91.

Define a set as **selfish** if it contains a number equal to its size. A set is **minimal selfish** if it is selfish and has no proper selfish subsets. Find the number of subsets of the set $ S = \{1, 2, 3, \cdots, n\} $, including $ \emptyset $.

92.

(1) Find the average size of subsets.

93.

(2) For $ s \subseteq S $, define $ f(s) = |s| $. Find the average value of $ f $.

94.

(3) $g(s) = \sum a \in sa$. Find the average value of $g$.

95.

(4) $h(s) = \max_{a \in s}(s)$. Find the average value of $h$.

96.

$X$ and $Y$ are two different whole numbers greater than 1. Their sum is not greater than 100, and $Y$ is greater than $X$. $S$ and $P$ are two mathematicians (and consequently perfect logicians); $S$ knows the sum $X + Y$ and $P$ knows the product $X \times Y$. Both $S$ and $P$ know all the information in this paragraph.

97.

The following conversation occurs (both participants are telling the truth):

98.

- $S$ says: "P does not know $X$ and $Y$." - $P$ says: "Now I know $X$ and $Y$." - $S$ says: "Now I also know $X$ and $Y$."

99.

What are $X$ and $Y$?

100.

(Sidney Sussex) Given that $a_0 = 0$, $a_1 = 1$, $a_2 = 1$, $a_3 = 2$ (Fibonacci Sequence), $S_n = a_0 + a_1 + a_2 + a_3 + \cdots$. Find an approximate form for $S_n$. Can any natural number $N$ be written as some sum of the Fibonacci numbers? (Numbers cannot be used repeatedly.)

5. 代数 (Algebra) (100题)

How many solutions are there to the equation $|x| + |x - 1| = 0$?

How many solutions are there to the equation $|27x^2 - 48| + |6x^2 - 5x - 4| = 0$?

What are the solutions of the equation $|\sin(2x)| + |\cos(0.5x)| = 0$?

Do you know a solution to the equation $5^x = 4^x + 3^x$? Are there any more? Prove it.

Let $h(x) = x^3 + ax$, where $a$ is a constant. When will an inverse to $h(x)$ exist for all $x$?

Show $(x - a)^2 - (x - b)^2 = 0$ has no real roots if $a$ does not equal $b$ in as many ways as you can.

Hence show:

(i) $(x - a)^3 + (x - b)^3 = 0$ has 1 real root.
(ii) $(x - a)^4 + (x - b)^4 = 0$ has no real roots if $a$ does not equal $b$.
(iii) $(x - a)^4 + (x - b)^4 = (b - a)^4$ has 2 real roots if $a$ does not equal $b$.

Where does $y = x^{x^{x^x}}$ exist (i.e. what is its domain)?

Given that $(\sqrt[3]{4} - \sqrt[3]{2})(\sqrt[3]{a} + \sqrt[3]{b} + \sqrt[3]{c}) = 2$, what does $abc$ equal?

10.

Find the value of $\sqrt{5 - 3\sqrt{2}}$

11.

Given \[ x = \sqrt{3\sqrt{2\sqrt{3\sqrt{2\sqrt{3\sqrt{2\cdots}}}}}} \] Find $x$.

12.

Find approximate roots: (i) for $ax^5 - x + 1 = 0$ (ii) for $ax^3 - x^2 - x + 1 = 0$

13.

(a) $x^n + 1 = 0$, $n$ is an odd integer, find a root of this equation.
(b) Can you factorise $x^n + 1$?
(c) $x^n + 1 = 0$, $n$ is an even integer, find a root.
(d) Given that $2^n + 1$ is prime, $n$ is an integer, prove that $n$ should be $2^k$, where $k$ is an integer.

14.

15.

Use algebraic techniques to determine whether the following equation has any real solutions: \[ 72 \]

16.

$ x^4 + 2x^3 + 3x^2 + 2x + 1 = 0 $

17.

If three positive real numbers $ a, b, c $ satisfy the following equations, show that at least one of them must be 1 and hence deduce all solutions: \[ abc = 1 \quad \text{and} \quad a + b + c = \frac{1}{a} + \frac{1}{b} + \frac{1}{c} \]

18.

Show that no three real numbers $ a, b, c $ satisfy the equations \[ a + b + c = 0 = \frac{1}{a} + \frac{1}{b} + \frac{1}{c}. \]

19.

Prove that if $ a, b, c $ are all odd then the quadratic equation $ ax^2 + bx + c = 0 $ cannot have rational roots.

20.

Consider the cubic curve given by the equation \[ y = ax^3 + bx^2 + cx + d \] find conditions on $ a, b, c, d $ which ensure the curve has a local maximum and a local minimum. Under these conditions, show that the curve has a point of inflection midway between the turning points.

21.

Find the sum of the coefficients of the polynomial obtained after expanding and collecting terms of the product \[ (1 - 3x + 3x^2 - 5x^3 + 5x^4)(1 + 3x - 3x^2 + 5x^3 - 5x^4) \]

22.

Find a polynomial with integer coefficients whose roots include $ \sqrt{2} + \sqrt{3} $.

23.

Prove that in the product \[ (1 - x + x^2 - x^3 + \cdots - x^{99} + x^{100})(1 + x + x^2 + \cdots + x^{99} + x^{100}) \] after multiplying out and collecting terms, there does not appear a term in $ x $ of odd degree.

24.

Determine $ m $, an integer, so that the equation \[ x^4 - (3m + 2)x^2 + m^2 = 0 \] has four real solutions for $ x $ that form an arithmetic progression.

25.

If the rational quantity $ \frac{p}{q} $ (in lowest terms) is a root of \[ f(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0 \] prove that $ p \mid a_0 $ and $ q \mid a_n $.

26.

Hence show that the $ n $th root of an integer is either an integer itself or irrational.

27.

Show that if four distinct points of the curve $ y = 2x^4 + 7x^3 + 3x - 5 $ are collinear then their average $ x $-coordinate is some constant $ k $. Find $ k $.

28.

Let $ P(x, y) $ be a polynomial in $ x $ and $ y $ such that: $ P(x, y) \equiv P(y, x) $ and $ (x - y) $ is a factor of $ P(x, y) $. Deduce that $ (x - y)^2 $ is a factor of $ P(x, y) $.

29.

Solve: $ x^4 + 3x^3 - 2x^2 - 3x + 1 = 0 $

30.

Sketch $ f(x) = x^5 + 2x^3 + x + 7 $ and $ g(x) = 12 - (x - 1)^4 $ on the same graph. Can you find any points of intersection?

31.

What are the conditions for which a cubic equation with real coefficients has two, one or no real solutions?

32.

Given $ x^2 - 2x + 2 $ has roots $ \alpha $ and $ \beta $, find a quadratic with roots $ \alpha^2 $ and $ \beta^2 $ without calculating $ \alpha $ and $ \beta $.

33.

Find the minimum value of the expression $ |x - 1| + |x - 2| + |x - 4| + |x - 6| $.

34.

By considering the inequality \[ \int_0^t (f(x) + \mu g(x))^2 \, dx \geq 0 \] where $ \mu $ is a constant, prove that, for all functions $ f(x) $ and $ g(x) $: \[ \left( \int_0^t f(x)g(x)\,dx \right)^2 \leq \left( \int_0^t (f(x))^2\,dx \right) \left( \int_0^t (g(x))^2\,dx \right) \quad \text{(Cauchy-Schwarz Inequality)} \] Hence show that \[ \int_0^1 (1 + x^5)^{\frac{1}{2}} \, dx \leq \sqrt{\frac{7}{6}} \]

35.

What is the shortest distance from the point $ A(3,1) $ to the curve with equation $ y = x^2 + 1 $?

36.

What is the shortest distance from the line $ y = x $ and the curve $ y = x^2 + 1 $?

37.

What is the shortest distance between the two curves $ y = x^2 + 1 $ and $ x = y^2 + 1 $?

38.

Prove that for any positive integer $ n $ and any real number $ x $, $ \left\lfloor \frac{[nx]}{n} \right\rfloor = [x] $ where $ [z] $ denotes the largest integer value less than or equal to $ z $.

39.

Prove $ \frac{a}{b} + \frac{b}{a} \geq 2 $ for positive $ a, b $.

40.

Which is larger $ (8!)^{1/8} $ or $ (9!)^{1/9} $?

41.

There is a set of numbers whose sum is equal to the sum of the elements squared. What’s bigger: the sum of the cubes or the sum of the fourth powers?

42.

Prove that arithmetic mean $ \geq $ geometric mean.

43.

Which is the quicker round trip by plane: no wind or a constant head wind (in one direction)?

44.

$ x \geq -1, n = 0, 1, 2, 3 \ldots $, Prove the following inequality is correct: \[ (1 + x)^n \geq 1 + nx \]

45.

Given there are 3 points inscribed by the circle, to maximize the area how the 3 points will be located? How many will there be for this area?

46.

Given $ x_1 + x_2 + \cdots + x_n = 0 $, and for any $ i $, $ |x_i| < 1 $. Now rearranged these $ n $ terms, does there exist $ j $ such that \[ |x_1 + x_2 + \cdots + x_j| < 1 \] is possible?

47.

Given that $ 100 = a_1 + a_2 + a_3 + \cdots + a_n $, where $ a_i > 0 $. Find the maximum value of $ a_1 a_2 a_3 \cdots a_n $.

48.

For each non-empty subset of integers $ (1, 2, 3, \ldots, n) $, consider the reciprocal of the product of the elements. Let $ S_n $ denote the sum of these reciprocals. Conjecture and prove a formula for $ S_n $.

49.

The lengths of the sides of a triangle are in geometric progression with common ratio $ r $. Prove that \[ \frac{2}{1 + \sqrt{5}} < r < \frac{1 + \sqrt{5}}{2} \]

50.

Simplify \[ 1^2 - 2^2 + 3^2 - 4^2 + \cdots + (2n - 1)^2 - (2n)^2 \]

51.

Find \[ 21^2 - 22^2 + 23^2 - 24^2 + \cdots + 39^2 - 40^2 \]

52.

If the ratio of consecutive Fibonacci numbers approaches a limit, what must this limit be?

53.

Given that \[ \sum_{r=1}^\infty \frac{1}{r^2} = \frac{\pi^2}{6} \] Find the exact value of \[ \sum_{r=1}^\infty \frac{1}{(2r - 1)^2} \]

54.

Find \[ \lim_{n \to \infty} \left( \frac{1}{n} + \frac{(n-1)^2}{n^3} + \frac{(n-2)^2}{n^3} + \cdots + \frac{1}{n^3} \right) \]

55.

Evaluate \[ \prod_{n=2}^\infty \left( 1 - \frac{1}{n^2} \right) \]

56.

Conjecture and prove a formula for \[ 1 \times 1! + 2 \times 2! + 3 \times 3! + \cdots + n \times n! \]

57.

Find the value of \[ 75 \]

58.

$ 1 + \frac{1}{1 + \frac{1}{1 + \cdots}} $

59.

Is the series \[ 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \cdots \] divergent?

60.

How about the series \[ 1 + \frac{1}{4} + \frac{1}{9} + \frac{1}{16} + \cdots \]

61.

Sketch on separate axes $ y = \frac{1}{x} $ and $ y = \frac{1}{x^2} $, considering your sketches and by using integration justify your claims.

62.

Consider the sequence $ 0, 1, 1, 2, 2, \ldots, r, r, r+1, r+1, \ldots $, deduce the sum of the first $ n $ terms $ S(n) $. Prove that \[ S(s + t) - S(s - t) = st \] where $ s $ and $ t $ are positive integers and $ s > t $.

63.

Prove that for $ n $ a positive integer, \[ 1 + \frac{1}{1!} + \frac{1}{2!} + \cdots + \frac{1}{n!} < 3 \]

64.

Find an approximation to \[ 1^4 + 2^4 + 3^4 + \cdots + 100^4 \]

65.

Prove that \[ 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \cdots \] equals infinity.

66.

If every term in a sequence $ S $ is defined by the sum of every item before it, give a formula for the $ n $th term.

67.

Investigate $ x_{n+1} = kx_n(1 - x_n) $ for different values of $ k $. Investigate limiting behaviour, if any. (Use a computer.)

68.

If $ u_n \to 0 $ as $ n \to \infty $, does $ \sum u_n $ converge? Proof or counterexample.

69.

Prove that \[ 1^5 + 2^5 + \cdots + n^5 = \frac{1}{12}n^2(n+1)^2(2n^2 + 2n - 1) \]

70.

Evaluate the limit of \[ \frac{n + n^2 + \cdots + n^n}{1^n + 2^n + \cdots + n^n} \] as $ n $ goes to infinity.

71.

A grid of size $ 2 \times n $ is covered with $ 2 \times 1 $ dominoes. How many ways are there to do this? What about a grid of size $ 3 \times n $ and $ 3 \times 1 $ dominoes?

72.

Given that $ a_0 = 0, a_1 = 1, a_2 = 1, a_3 = 2 $ (Fibonacci Sequence), $ S_n = a_0 + a_1 + a_2 + a_3 + \cdots $, find an approximate form for $ S_n $. Can any natural number $ N $ be written as some sum of the Fibonacci numbers? (Numbers cannot be used repeatedly.)

73.

They introduced a sequence called Fibonacci sequence. $0, 1, 1, 2, 3, 5, \ldots$ $f(0) = 0$, $f(1) = 1$

74.

Find an expression for $f(n)$ when $(n \geq 2)$

75.

(a)
What is a series, given a series.
What is the convergence of a series?
Are the following series convergent: [(i)]
$1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \ldots$
$1, 2, 4, 8, \ldots$
$1, 1, 1, 1, \ldots$
$1, -1, 1, -1, 1, \ldots$
$a_n = \dfrac{5n^2 + 2n + 3}{n+1}$
$a_n = \sin(n)$

76.

Given that $a_1 = 4$, $a_2 = 484$, $a_3 = 48484$, $b_1 = 8$, $b_2 = 848$, $b_3 = 84848$.

(a)
Find the general expression of $a_n$ and $b_n$.
Show that $4b_n - 7a_n = 4$.

77.

What is the value of $\displaystyle \sum_{k=0}^{n} \binom{n}{k} (-1)^k$?

78.

$(6 + \sqrt{37})^{20} \approx 4398704651\ldots$ with the digits after decimal point $.9999999999437569\ldots$ Can you show why this number is so close to an integer?

79.

Pascal’s triangle (prove that every number in the triangle is the sum of the two above it)

80.

Prove that \[ \sum_{k=0}^{n} \binom{n}{k} = 2^n, \quad \sum_{k=0}^{n} (-1)^k \binom{n}{k} = 0 \] and \[ \sum_{k=0}^{n} \binom{n}{k}^2 = \binom{2n}{n} \]

81.

Solve $a^b = b^a$ for all real $a$ and $b$. What does this have to do with $y = \dfrac{\ln x}{x}$?

82.

Which will be larger as $n \to \infty$: $2^{2^{2n}}$ or $100^{100n}$?

83.

Given that $8 < \pi^2 < 10$, show that \[ \frac{1}{\log_2(\pi)} + \frac{1}{\log_5(\pi)} > 2 \quad \text{and} \quad \frac{1}{\log_2(\pi)} + \frac{1}{\log_\pi(2)} > 2 \]

84.

If a natural number $ n $ has $ N $ digits, how many digits can $ n^2 $ have? What about $ n^n $? How would you write a formula for the number of digits of $ n $?

85.

What is the domain and range of the functions: $ f(x) = \ln(x) $, $ ff(x) $, and $ ff(x) $? What about $ f^n(x) $?

86.

Find explicit expressions for $ \sinh^{-1}(x) $, $ \cosh^{-1}(x) $, and $ \tanh^{-1}(x) $.

87.

If $ 2\log(x - 2y) = \log(x) + \log(y) $, find $ \frac{x}{y} $.

88.

Evaluate, without the use of a calculator, $ (\log_3 169)(\log_{13} 243) $.

89.

Which is bigger: $ 2^x $ or $ x^2 $?

90.

Which is greater: $ e^\pi $ or $ \pi^e $?

91.

Prove that for $ n \in \mathbb{Z}, n > 2 $, $ n^{(n+1)} > (n+1)^n $.

92.

What is larger as $ n \to \infty $: $ 2^{2^{2n}} $ or $ 100^{100n} $?

93.

Solve $ e^{2x} - 5e^x + 6 \geq 0 $.

94.

Given $ f(x) = \ln(\ln(\ln(x))) $, find the range of $ x $ when $ f(x) > 0 $ and $ f(x) < 0 $.

95.

Find the exact value of $ \cos^2(1^\circ) + \cos^2(2^\circ) + \cos^2(3^\circ) + \ldots + \cos^2(89^\circ) $.

96.

Show that $ \cos(n\theta) = f_n(\cos(\theta)) $ for polynomials $ f_n(x) $ satisfying $ f_{n+1}(x) = 2x f_n(x) - f_{n-1}(x) $. Find all the roots of $ f_2(x) + f_3(x) = 0 $, and write them in the form $ \cos(\phi) $ for suitable $ \phi $.

97.

Find the smallest $ a > 1 $ such that $ \frac{a + \sin(x)}{a + \sin(y)} \leq e^{y - x} $ for all $ x \leq y $.

98.

Where in the plane is $ \sin^2(x) + \cos^2(y) = 1 $?

99.

Prove $\sin x = x \prod_{r=1}^{\infty} \cos\left(\frac{x}{2^r}\right)$

100.

What is the square root of $i$?

6. 微积分 (Calculus) (100题)

Differentiate $ y = x^{\sin x} $

What do you know about differentiation?

Looking for $ \lim_{\delta x \to 0} f'(x) = \frac{f(x+\delta x) - f(x)}{\delta x} $.

Then, use this to work out: $ \frac{d}{dx}(x^2) $, $ \frac{d}{dx}(x^3) $ and $ \frac{d}{dx}(x^n) $

Differentiate $ x^x $

Differentiate $ x^{x^x} $

Differentiate $ \frac{1}{x + \frac{1}{x + \frac{1}{x}}} $ w.r.t. $ x $.

What are $ \left. \frac{d^n}{dx^n} \left( e^{-\frac{1}{x^2}} \right) \right|_{x=0} $ $ \forall n \in \mathbb{N} $? What does this mean for its Maclaurin series?

Differentiate $ x^x $ and $ (x^{0.5})^{(x^{0.5})} $.

10.

You are given that $ y = t^t $ and $ x = \cos t $. What is the value of $ dy/dx $?

11.

Differentiate $ y = x $ with respect to $ x^2 $?

12.

Taylor series $ \cos x = \sum a_i \times x^j $ and use this to show the coefficient for the first three terms are $ \cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots $

13.

Differentiate $ x \cos \frac{3}{x} $

14.

The curve equation: $ \sqrt[8]{|x|} + \sqrt[8]{|y|} = 1 $, when does the tangent of curve perpendicular to $ x $-axis?

15.

Differentiate $ y = \log \left( \sin \left( e^{x^2} \right) \right) $.

16.

Given $ a(x) = \left(1 + \frac{1}{x}\right)^x $

17.

(a) Find $ a'(x) $ (b) Show that $ 1 + \frac{1}{x} > e^{\frac{1}{1+x}} $ (c) Then deduce that $ \log \left(1 + \frac{1}{x}\right) > \frac{1}{1+x} $ (d) Is $ a(x) $ an increasing or decreasing function?

18.

Suppose that $ f(0) = 0 $ and that, for $ x \neq 0 $, $ 0 < \frac{f(x)}{x} < 1 $. Show that \[ -\frac{1}{2} < \int_{-1}^{1} f(x)\,dx < \frac{1}{2} \] How does the above inequality change if $ 0 < \frac{f(x)}{x^2} < 1 $ instead?

19.

Find $ \frac{dy}{dx} $ when $ y = \int_0^x t^8 e^t\,dt $.

20.

Let \[ A = \int \frac{\sin(x)}{\sin(x) + \cos(x)}\,dx \quad \text{and} \quad B = \int \frac{\cos(x)}{\sin(x) + \cos(x)}\,dx \] find $ A $ and $ B $.

21.

Integrate $ \sin^4(x)\cos(x) $ and $ \sin^6(x)\cos^3(x) $, in general when will this method work?

22.

Evaluate $ I = \int \frac{1}{x^n + x}\,dx $.

23.

Prove that for a continuous function \[ \int_b^a f(x)\,dx = \int_b^a f(a + b - x)\,dx \] Hence evaluate \[ I = \int_4^8 \frac{\ln(9 - x)}{\ln(9 - x) + \ln(x - 3)}\,dx \quad \text{and} \quad J = \int_0^{\frac{\pi}{2}} \frac{\sin^{2000}(\theta)}{\sin^{2000}(\theta) + \cos^{2000}(\theta)}\,d\theta \]

24.

Evaluate $ I = \int_0^1 \frac{1}{\sqrt{x + 3\sqrt{x}}}\,dx $.

25.

By considering the graph of the function $ f(x) = x^{-s} $, show that \[ \frac{1}{s - 1} < 1 + 2^{-s} + 3^{-s} + \cdots < \frac{s}{s - 1} \]

26.

Evaluate \[ I = \int \frac{1}{1 - \sin(x)}\,dx, \quad J = \int e^x \sin(x)\,dx, \quad K = \int \sqrt{e^{2x} + 1}\,dx \]

27.

Which of the following numbers is bigger and why? \[ \int_0^1 \sqrt[7]{1 - x^4}\,dx \quad \text{or} \quad \int_0^1 \sqrt[4]{1 - x^7}\,dx \]

28.

Show that \[ \int_{\frac{\pi}{2}}^{\pi} \frac{x \sin(x)}{1 + \cos^2(x)}\,dx = \int_0^{\frac{\pi}{2}} \frac{(\pi - x)\sin(x)}{1 + \cos^2(x)}\,dx \] Hence find \[ I = \int_0^{\pi} \frac{x \sin(x)}{1 + \cos^2(x)}\,dx \]

29.

Show that \[ \int_0^1 \frac{\sqrt{x^2}}{\sqrt{1 - x^2}}\,dx = \int_0^1 \frac{1}{\sqrt{1 - x^2}}\,dx \]

30.

Hence find

31.

\[ I = \int_0^1 \frac{x^2}{\sqrt{1 - x^2}} \, dx \]

32.

\[ \int_0^2 \frac{1}{(1 - x)^2} \, dx \]

33.

Integrate $\cos^2(x)$ and $\cos^3(x)$.

34.

Integrate $y = \ln x$.

35.

Prove that $1 + \frac{1}{2} + \frac{1}{3} + \ldots + \frac{1}{1000} < 10$.

36.

Integrate $y = \tan x$.

37.

Integrate and differentiate $x \ln(x)$.

38.

Integrate $y = \frac{1}{(1 - \ln x)}$.

39.

Integrate $y = 1/(1 + x^2)$.

40.

Integrate $y = x^{-2}$ between limits of $1$ and $-1$. Draw the graph.

41.

Why is it $-2$ and not infinity, as it appears to be on the graph?

42.

Integrate $e^x x^2$ between limits of $0$ and $1$. Draw that graph.

43.

Integrate $y = 1/(x + x^3)$, $y = 1/(1 + x^3)$, $y = 1/(1 + x^n)$.

44.

Integrate $1/(9 + x^2)$.

45.

Integrate $y = x \sin(2x)$.

46.

Integrate $1/(1 + \sin x)$.

47.

Find $\displaystyle \int \frac{dx}{x \ln x}$.

48.

Find $\displaystyle \int x \sin(x^2) \, dx$.

49.

Find $\int_0^{2\pi} |\sin^n x + \cos^n x| \, dx$ for $n = 1, 2, \cdots$.

50.

Sketch $y = \ln(\sin x)$ for $0 < x < \pi$ and $y = \ln(\cos x)$ for $0 < x < \pi/2$. Evaluate the integral of $\ln(\sin x)$ from $0$ to $\pi/2$.

51.

Integrate from $0$ to infinity the following: $\int x e^{-x^2} \, dx$ and $\int x^3 e^{-x^2} \, dx$.

52.

Draw the functions $\frac{1}{x}$ and $\frac{1}{\sqrt{x}}$ on one graph, and then find the integral for both between $0$ and $1$. Which one is finite/infinite? For the general case $x^{-\alpha}$, for what value of $\alpha$ is the integral between $0$ and $1$ finite and infinite?

53.

Derive a reduction formula for the integral: $I(n) = \int_0^\infty x^n e^{-x^2} \, dx$. Hence, evaluate $I(n)$, what have you learned about the standard normal distribution?

54.

Set \[ I = \int_0^a \frac{\cos x}{\cos x + \sin x} \, dx, \quad J = \int_0^a \frac{\sin x}{\cos x + \sin x} \, dx. \] Prove that in the range $0 < a < \frac{3\pi}{4}$, $2I = a + \ln(\cos a + \sin a)$.

55.

Evaluate: \[ \int_{-\frac{\pi}{2}}^{\frac{(2n+1)\pi}{2}} x^2 \cos x \, dx, \] \[ \int_{-\frac{\pi}{2}}^{\frac{(2n+1)\pi}{2}} x^2 |\cos x| \, dx. \]

56.

Solve the differential equation for $\frac{dy}{dx} = ky$ subject to the initial condition $x = 0, y = 1$ and $k > 0$. Sketch the solution to the differential equation \[ \frac{dy}{dx} = ky\left(1 - \frac{y}{M}\right) \] where $M$ is a large constant and the same initial conditions apply (without directly finding $y$).

57.

Find $f(x)$ if \[ \int_0^x f(t) \, dt = 3f(x) + k \] where $k$ is a constant.

58.

Draw the graph of $\frac{dy}{dx} = \frac{y - 3}{y^2 + x^2}$ given that it passes through $(0, 1)$.

59.

Solve $\frac{dn}{dt} = -\lambda n$ then give $n$ in terms of $-\lambda$. If $n = 1$ and $\lambda = 0$, find the constant of integration.

60.

Find a general expression for $f(x)$ when: \[ f''(x) = 0, \] \[ f''(x) = f(x), \] \[ f''(x) = 4f(x). \]

61.

Solve the differential equation $\frac{dy}{dx} = ky$ for $x > 0$ subject to the initial condition $x = 0, y = 1$ and $k > 0$. Sketch the solution to the differential equation $\frac{dy}{dx} = ky\left(1 - \frac{y}{M}\right)$ where $M$ is a large constant and the same initial conditions apply (without directly finding $y$).

62.

Find $f(x)$ if $\int_0^x f(t)\,dt = 3f(x) + k$, where $k$ is a constant.

63.

Is the series $1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \cdots$ divergent? How about the series $1 + \frac{1}{4} + \frac{1}{9} + \frac{1}{16} + \cdots$?

64.

Sketch on separate axes $y = \frac{1}{x}$ and $y = \frac{1}{x^2}$, considering your sketches and by using integration justify your claims.

65.

By considering the inequality $\int_0^t (f(x) + \mu g(x))^2 dx \geq 0$, where $\mu$ is a constant, prove that, for all functions $f(x)$ and $g(x)$: \[ \left(\int_0^t f(x)g(x)\,dx\right)^2 \leq \left(\int_0^t (f(x))^2 dx\right)\left(\int_0^t (g(x))^2 dx\right) \] (Cauchy-Schwarz Inequality)

66.

Hence show that $\int_0^1 (1 + x^5)^{\frac{1}{2}} dx \leq \sqrt{\frac{7}{6}}$.

67.

Let $A = \int \frac{\sin(x)}{\sin(x) + \cos(x)} dx$ and $B = \int \frac{\cos(x)}{\sin(x) + \cos(x)} dx$, find $A$ and $B$.

68.

Evaluate $I = \int \frac{1}{x^n + x} dx$

69.

Prove that for a continuous function $\int_a^b f(x)\,dx = \int_a^b f(a + b - x)\,dx$.

70.

Hence evaluate i) $I = \int_4^8 \frac{\ln(9 - x)}{\ln(9 - x) + \ln(x - 3)} dx$ ii) $J = \int_0^{\frac{\pi}{2}} \frac{\sin^{2000}(\theta)}{\sin^{2000}(\theta) + \cos^{2000}(\theta)} d\theta$

71.

Evaluate $I = \int_0^1 \frac{1}{\sqrt{x} + \sqrt[3]{x}} dx$.

72.

Evaluate i) $I = \int \frac{1}{1 - \sin(x)} dx$ ii) $J = \int e^x \sin(x)\,dx$ iii) $K = \int \sqrt{e^{2x} + 1}\,dx$

73.

Which of the following numbers is bigger and why: $ I = \int_0^1 \sqrt[7]{1 - x^4} \, dx $ or $ \int_0^1 \sqrt[4]{1 - x^7} \, dx $?

74.

Show that $ \int_{\frac{\pi}{2}}^{\pi} \frac{x \sin(x)}{1 + \cos^2(x)} \, dx = \int_0^{\frac{\pi}{2}} \frac{(\pi - x) \sin(x)}{1 + \cos^2(x)} \, dx $.

75.

Hence find $ I = \int_0^{\pi} \frac{x \sin(x)}{1 + \cos^2(x)} \, dx $.

76.

Show that $ \int_0^1 \frac{x^2}{\sqrt{1 - x^2}} \, dx = \int_0^1 \sqrt{1 - x^2} \, dx $.

77.

Hence find $ I = \int_0^1 \frac{x^2}{\sqrt{1 - x^2}} \, dx $.

78.

What is the shortest distance from the point $ A(3,1) $ to the curve with equation $ y = x^2 + 1 $?

79.

What is the shortest distance from the line $ y = x $ and the curve $ y = x^2 + 1 $?

80.

What is the shortest distance between the two curves $ y = x^2 + 1 $ and $ x = y^2 + 1 $?

81.

\[ \int \cos x \cdot \sin^2 x \, dx \]

82.

\[ LHS = \int \sin^2 x \, d(\sin x) = \frac{1}{3} \sin^3 x + c \]

83.

\[ \int \sin^m x \cos^n x \, dx, \, m,n \in \mathbb{N}, \text{ for what condition of } m,n, \text{ the integral is easy to solve?} \]

84.

\[ 241n(n+1)(n+2)(n+3) \]

85.

\[ k! \mid n(n+1)\cdots(n+k-1) \]

86.

Is $ \tan 1^\circ $ irrational?

87.

Is $ \cos 1^\circ $ irrational?

88.

\[ \int \tan^{-1} x \, dx \]

89.

Find $ \left[ \frac{f(x)}{g(x)} \right]' $

90.

\[ P(x=1) = \frac{1}{f(s)}, \, f(s) = \frac{1}{\prod_p \left(1 - \frac{1}{p^s}\right)} \]

91.

\[ \int_y^y \int_x^x dx \, dy \]

92.

Find $\int_{x=0}^{x=1} \int_{y=0}^{y=3-3x} dy\,dx$. Write the expression for the area $R$ in unit circle.

93.

Find $\int_{x=0}^{x=4} \int_{y=0}^{y=\sqrt{x}} dy\,dx$.

94.

Find the area bounded by $y^2 = 4x$ and $2x + y = 4$.

95.

$y = 2x^4 + 7x^3 + 3x - 5$, four distinct points on $y$ are collinear, find the average of $x$-coordinates.

96.

$$ \int_0^{\frac{\pi}{2}} \cos^2 x\,dx $$

97.

$$ \int_0^{\frac{\pi}{2}} \cos^3 x\,dx $$

98.

$$ \int_0^{\frac{\pi}{2}} \ln(\cos x)\,dx $$

99.

Evaluate $I = \int \frac{1}{1 - \sin x}\,dx$.

100.

Which of the following numbers is bigger and why: $I = \int_0^1 (1 - x^7)^{\frac{1}{4}}\,dx$ or $J = \int_0^1 (1 - x^4)^{\frac{1}{7}}\,dx$?

7. 数学建模 (Model) (100题)

Imagine you are in a lift travelling from the bottom of a building to the top.

Sketch the velocity time graph for this situation. Imagine a person in the lift standing on scales, sketch a graph of the reading on the scales.

What forces are acting?

Do you notice anything interesting about the bumps?

Two trains are moving, so that the first train with mass $m$ is moving at $v \text{ ms}^{-1}$, and the second train with mass $M$ is stationary. The first train collides with and sticks to the second train. What is the speed of the train after the collision?

What other forms of energy is the Kinetic Energy converted into on impact? Work out the energy loss and simplify it.

Particle moving at $v \text{ ms}^{-1}$, with $a \text{ ms}^{-2}$ acceleration, and time $t$, travelling from rest.

Please give an equation linking it all. Use this equation with $u \neq 0$ to find the distance.

At which points on a roller coaster are you most likely to fall off?

10.

If a cannon is pointed straight at a monkey in a tree, and the monkey lets go and falls towards the ground at the same instant the cannon is fired, will the monkey be hit? Describe any assumptions you make.

11.

What makes a tennis ball spin as it's travelling through the air?

12.

Draw the graph of which the centre of the ladder draws during sliding.

13.

Find the expression of the area that the ladder creates when sliding down.

14.

$x$ and $y$ symmetry

15.

16.

17.

There is a circular lake with radius $50 \text{ m}$. There is a tree at the centre and a tree at the side. How to

18.

use a rope length 101 m to climb to the tree at the centre from the tree at the side.

19.

Jack and Sam live at opposite ends of the same street. Jack had to deliver a parcel to Sam’s house and Sam had to deliver one to Jack’s house. They started at the same moment, each walked a constant speed and returned home immediately after leaving the parcel at its destination. They met first time at the distance of $a$ metres from Jack’s house and the second time $b$ metres from Sam’s house.

(a)
How long is the street?
If $a = 300$ and $b = 400$, who walks faster?

20.

Tom, Dick and Harry travel together. Dick and Harry are good hikers; each walks $p$ miles per hour. Tom has a bad foot and drives a small car in which two people can ride, but not three; the car covers $c$ miles per hour. The three friends adopted the following scheme:

21.

They start together, Harry rides in the car with Tom, Dick walks. After a while, Tom drops Harry who walks on; Tom returns to pick up Dick and then Tom and Dick ride in the car till they overtake Harry. At this point they change and Harry rides with Tom while Dick walks just as they started. The whole procedure is repeated as often as necessary.

(a)
How much distance do they cover per hour?
For what fraction of the time taken to travel the journey does the car carry only one person?
Check the extreme cases $p = 0$ and $p = c$.

22.

(St Hilda’s) With $n$ sets of data, $x_1, x_2, \ldots, x_n, y_1, y_2, \ldots, y_n$. Find the line of best fit for the linear regression model $y_i = a + bx_i$.

23.

(Lady Margaret Hall) Find the equation of the curve of the ladder as it falls (Consider $\theta$ fixed, not the mid-point). Sketch the solution of curve.

24.

How long does a mirror have to be for you to see your whole body?

25.

A body with mass $m$ is falling towards earth with speed $v$. It has a drag force equal to $kv$. Set up a differential equation and solve it for $v$.

26.

A body with mass $'m'$ is falling towards earth with speed $v$. It has a drag force equal to $kv$. Set up a differential equation and solve it for $v$.

27.

You have a 3 litre jug and a 5 litre jug. Make 4 litres.

28.

Consider two identical, frictionless slopes, down which we send two identical particles. If each particle starts the same height up each slope, but one rolls whereas the other simply translates down the slope, which particle will reach the bottom first?

29.

I am an oil baron in the desert and I need to deliver oil to four different towns which happen to lie on a straight line. In order to deliver the correct amounts to each town, I must visit each town in turn, returning to my warehouse in between each visit. Where should I position my warehouse in order to drive the shortest distance possible? Roads are no problem since I have a friend who is a sheikh and will build me as many roads as I like for free.

30.

I drove to this interview at 50 kmph and will drive back at 30 kmph because of the traffic. What is my average speed?

31.

Derive the formula for the volume of a sphere.

32.

Imagine a ladder leaning against a vertical wall with its feet on the ground. The middle rung of the ladder has been painted a different colour on the side, so that we can see it when we look at the ladder from the side on. What shape does that middle rung trace out as the ladder falls to the floor?

33.

If I go from $A$ to $B$ at 40 mph, and back from $B$ to $A$ at 60 mph, assuming there is no time taken in the turn, is my average speed more or less than 50 mph?

34.

35.

A body with mass $m$ is falling towards Earth with speed $v$. It has a drag force equal to $kv$. Set up a differential equation and solve it for $v$.

36.

differential equation and solve it for $v$.

37.

Problem 63: Two trains starting 30 km apart and travelling towards each other. They meet after 20 minutes. If the faster train chases the slower train (starting 30 km apart), they meet after 50 minutes. How fast are the trains moving?

38.

6. A packing case is held on the side of a hill and given a kick down the hill. The hill makes an angle of $\theta$ to the horizontal, and the coefficient of friction between the packing case and the ground is $\mu$. What relationship between $\mu$ and $\theta$ guarantees that the packing case eventually comes to rest? Let gravitational acceleration be $g$. If the relationship above is satisfied, what must the initial speed of the packing case be to ensure that the distance it goes before stopping is $d$?

39.

8. One end of a rod of uniform density is attached to the ceiling in such a way that the rod can swing about freely with no resistance. The other end of the rod is held still so that it touches the ceiling as well. Then the second end is released. If the length of the rod is $l$ metres and gravitational acceleration is $g$ metres per second squared, how fast is the unattached end of the rod moving when the rod is first vertical?

40.

10. Consider a mass $m$ at position $x(t)$ on a rough horizontal table attached to the origin by a spring with constant $k$ (restoring force $-kx$) and with a dry friction force $f$ defined by \[ f = \begin{cases} F & \text{if } \dot{x} < 0 \\ -F \leq f \leq F & \text{if } \dot{x} = 0 \\ -F & \text{if } \dot{x} > 0. \end{cases} \] What is the range of $x$ where the mass can rest? Show that if the mass moves, the maximum excursion decreases by $2F/k$ per half cycle. Discuss the motion.

41.

10. A cylindrical spaceship of mass $M$ and cross-sectional area $A$ is coasting at constant velocity when it suddenly encounters a dust cloud. The captain is dismayed to find that the dust sticks to the spaceship. If the density of dust is $\rho$, how far does the ship travel before its velocity is reduced by half?

42.

注意这个图像我是用软件制作的，题目并没有提使图像

43.

Draw the graph of which the centre of the ladder draws during sliding. (Submitted by 2016 Oxford)

44.

Solution:

45.

Assume $M(x,y)$ is a mid-point of the ladder and $c$ the length of the ladder, and $A(2x,0)$, $B(0,2y)$. There is the Pythagorean theorem $(2x)^2 + (2y)^2 = c^2$, so $x^2 + y^2 = \left(\frac{c}{2}\right)^2$. This is the equation of a circle.

46.

Result: The centre of the ladder is moving along a circle.

47.

The Monkey and the Hunter

48.

102. If a cannon is pointed straight at a monkey in a tree, and the monkey lets go and falls towards the ground at the same instant the cannon is fired, will the monkey be hit? Describe any assumptions you make. (Submitted by Oxford Applicant, The Student Room)

49.

50.

Assumptions: - No air resistance. - Constant gravitational acceleration $ g $. - The cannonball follows a parabolic trajectory under gravity. - The monkey and cannonball start moving simultaneously. - The initial velocity of the cannonball is sufficient to reach the monkey's initial position.

51.

Yes, the monkey will be hit. Both the cannonball and the monkey experience the same downward acceleration due to gravity. Since the cannon is aimed directly at the monkey's initial position, and both begin falling at the same time, they will remain aligned vertically throughout their motion. Thus, the cannonball will intersect the monkey's path at the point where it has fallen.

52.

1. Lady Margaret Hall

53.

Given a donut (torus), state the coordinate of the point on the surface. You will be given two angles, one for horizontal plane and another for vertical plane. You will also be given two distances, one for inner circle and another for outer circle. Derive a formula including two angles and two distances to express the location of that point.

54.

Let: - $ R $ be the distance from the center of the torus to the center of the tube (major radius), - $ r $ be the radius of the tube (minor radius), - $ \theta $ be the angle in the horizontal plane (azimuthal angle), - $ \phi $ be the angle in the vertical plane (poloidal angle).

55.

Then the parametric coordinates of a point on the torus are:

56.

\[ x = (R + r \cos \phi) \cos \theta \] \[ y = (R + r \cos \phi) \sin \theta \] \[ z = r \sin \phi \]

57.

Then derive a formula that doesn't contain angles.

58.

To eliminate angles, consider the following:

59.

From the equations: \[ x^2 + y^2 = (R + r \cos \phi)^2 \] \[ z = r \sin \phi \Rightarrow z^2 = r^2 \sin^2 \phi \]

60.

Now, \[ x^2 + y^2 = R^2 + 2Rr \cos \phi + r^2 \cos^2 \phi \] \[ x^2 + y^2 - R^2 = 2Rr \cos \phi + r^2 \cos^2 \phi \]

61.

Also, \[ \cos^2 \phi = 1 - \frac{z^2}{r^2} \]

62.

Substitute into the equation: \[ x^2 + y^2 - R^2 = 2Rr \cos \phi + r^2 \left(1 - \frac{z^2}{r^2}\right) = 2Rr \cos \phi + r^2 - z^2 \]

63.

But this still contains $ \cos \phi $. To fully eliminate angles, use:

64.

\[ (x^2 + y^2 - R^2)^2 + z^2 = r^2 (x^2 + y^2) \]

65.

This is the implicit equation of the torus without angles.

66.

The characteristic of positive integral that is divisible by 3.

67.

A positive integer is divisible by 3 if and only if the sum of its digits is divisible by 3.

68.

Guess the characteristic of positive integral that is divisible by 9 then state why.

69.

A positive integer is divisible by 9 if and only if the sum of its digits is divisible by 9.

70.

Why? Because any number can be written as: \[ N = a_n \cdot 10^n + a_{n-1} \cdot 10^{n-1} + \cdots + a_0 \] and since $ 10^k \equiv 1 \pmod{9} $, we have: \[ N \equiv a_n + a_{n-1} + \cdots + a_0 \pmod{9} \] Thus, $ N \equiv \text{sum of digits} \pmod{9} $, so divisibility by 9 depends on the digit sum.

71.

已知 $ V, \alpha $, how much should be $ \theta $ that the distance on the plane is greatest?

72.

hint : $ \because \alpha $ fixed, length max $ \to \alpha_{\max} $

73.

根据 $ x = \cdots, y \cdots $ 约掉 $ t $,

74.

$ x_{\max} \to X $ 式子 differentiation $ = 0 $,

75.

根据 2 倍角公式化简,

76.

最后得到 $ \theta $ 的关于 $ \alpha $ 式子.

77.

($ \theta $ 为角平分时 distance max)

78.

Let’s interpret this problem: Given initial speed $ V $ and angle $ \alpha $ between two directions (perhaps projectile launched at angle $ \theta $ relative to some axis), find $ \theta $ such that the horizontal range or distance on the plane is maximized.

79.

Assume: A projectile is launched with speed $ V $ at an angle $ \theta $ above the horizontal, and we want to maximize the horizontal distance traveled before hitting the ground.

80.

Standard range formula: \[ R = \frac{V^2 \sin(2\theta)}{g} \]

81.

Maximized when $ \sin(2\theta) = 1 $, i.e., $ 2\theta = 90^\circ \Rightarrow \theta = 45^\circ $.

82.

But here, the hint suggests $ \theta $ is related to $ \alpha $, possibly meaning the launch angle is constrained or there is a fixed angle $ \alpha $ between two vectors.

83.

Alternatively, suppose we are given a fixed total angle $ \alpha $, and we split it into two parts $ \theta $ and $ \alpha - \theta $, and want to maximize the distance.

84.

Or perhaps: Two projectiles launched at angles $ \theta $ and $ \alpha - \theta $, and we want to maximize the sum of ranges.

85.

But based on the hint: “$ \theta $ 为角平分时 distance max”, meaning maximum distance occurs when $ \theta $ is half of $ \alpha $, i.e., $ \theta = \frac{\alpha}{2} $.

86.

So likely, the setup is symmetric: e.g., launching from a point with total angular span $ \alpha $, and choosing $ \theta $ such that the trajectory reaches farthest.

87.

Suppose we have a projectile launched with speed $ V $, and the direction makes angle $ \theta $ with the horizontal, but constrained such that the total angle between two possible paths is $ \alpha $. Or more plausibly, the problem is about maximizing the horizontal distance when the launch angle is $ \theta $, and $ \alpha $ is a fixed parameter (e.g., angle of incline).

88.

But the hint says: “$ \alpha $ fixed, length max $ \to \alpha_{\max} $” — possibly typo.

89.

Another interpretation: Suppose we are given a fixed speed $ V $, and a fixed angle $ \alpha $ between the initial velocity vector and some reference, and we vary $ \theta $ (say, the elevation) to maximize horizontal distance.

90.

But the final conclusion is: $ \theta = \frac{\alpha}{2} $ gives maximum distance.

91.

So assume: We are to launch a projectile such that the angle between the initial velocity and the horizontal is $ \theta $, and we are told that the total angle available or constraint is $ \alpha $, and we want to maximize range.

92.

But standard result: maximum range at $ \theta = 45^\circ $, independent of $ \alpha $.

93.

Unless $ \alpha $ is not the launch angle, but something else.

94.

Perhaps: Given a fixed speed $ V $, and a fixed angle $ \alpha $ between the launch direction and a wall or slope, find $ \theta $ to maximize distance along the plane.

95.

But based on the Chinese text:

96.

“根据 $ x = \cdots, y \cdots $ 约掉 $ t $,”

97.

So eliminate $ t $ from parametric equations.

98.

Assume: \[ x = V \cos \theta \cdot t, \quad y = V \sin \theta \cdot t - \frac{1}{2} g t^2 \]

99.

Eliminate $ t $: $ t = \frac{x}{V \cos \theta} $

100.

Substitute: \[ y = x \tan \theta - \frac{g x^2}{2 V^2 \cos^2 \theta} \]

8. 综合问题 (General) (75题)

How many vertices and edges does a line segment have? A square? A cube? A tesseract?

Can you conjecture formulae for the number of edges and vertices of an $n$ dimensional hypercube?

Can you give the coordinates of the vertices of a tesseract (where 4 edges coincide with the coordinate axes)?

What would the longest length between two vertices be?

Prove that there are infinitely many primes.

Prove that there are infinitely many primes of the form $4n + 3$.

Show that $3 < \pi < 4$.

Is $\frac{1}{n+1} \binom{2n}{n}$ always integer valued when $n$ is a positive integer?

Prove infinity of primes, prove infinity of primes of form $4n + 1$.

10.

Define the term 'prime number'.

11.

Find method to find if a number is prime.

12.

Explain what integration is.

13.

Explain the problem and solution of the Bridges of Königsberg (Euler).

14.

Find the value of the infinite continued fraction $[1, 1, 1, 1, \dots]$.

15.

$A$ is a $2 \times 2$ matrix, investigate $e^A$.

16.

Look up Fermat's little theorem.

17.

Look up Carmichael numbers.

18.

What is the significance of prime numbers?

19.

Do you know where the multiplication sign came from?

20.

A twin prime is a prime number that is either 2 less or 2 more than another prime number. For example, 11 and 13.

21.

Show that the only triple prime (i.e.\ three primes that are 2 less or 2 more than each other) is $3, 5, 7$.

22.

If you could have half an hour with any mathematician past or present, who would it be?

23.

Prove that integers greater than 1 can be written in the form of multiplication of prime numbers.

24.

With $n$ sets of data, $x_1, x_2, \ldots, x_n, y_1, y_2, \ldots, y_n$. Find the line of best fit for the linear regression model \[ y_i = a + bx_i \]

25.

Set $F_n = 2^{2^n} + 1$, $F_{n+k} = 2^{2^{n+k}} + 1$. Prove that $F_n$ and $F_{n+k}$ are coprime. (Hint: $\frac{F_{n+k} - 2}{F_n}$ is an integer)

26.

Given: For sets $A$ and $B$, $A$ and $B$ are said to be similar if there exists a bijection $f$ such that for any two elements $x$ and $y$ in $A$, if $x < y$ then $f(x) < f(y)$.

27.

(i) Show that whether the following two sets are similar or not:

(a) $\mathbb{Z}$ and $\mathbb{N}$
(b) $\mathbb{Z}$ and $\mathbb{Q}$

28.

(ii) Does there exist a bounded set similar to $\mathbb{N}$?

29.

(iii) Does there exist a bounded set similar to $\mathbb{Z}$?

30.

(iv) Knowing that a bounded set of $0$ to $1$ and a bounded set of $1$ to $2$ are both similar to $\mathbb{N}$, does their intersection is similar to $\mathbb{N}$?

31.

What was the most beautiful proof in A-Level Mathematics?

32.

What do you think is beautiful in maths?

33.

Do you know what a hyperbolic function is?

34.

What are modular functions?

35.

What do you know about Fermat’s last theorem?

36.

What is the significance of Euler's equations?

37.

What is your favourite number?

38.

Why do we approximate many functions in maths as cosine?

39.

Why is there a shortage of mathematicians today?

40.

Which is more important: representing your college or completing your maths assignments?

41.

Is maths a universal language?

42.

Talk about an interesting piece of maths. Why mathematics?

43.

How many hours a week do I study?

44.

Which branch of maths do I find interesting?

45.

Tell me about binary searches. What about their efficiency?

46.

How do you understand Newton's Laws?

47.

If you have a sorted array of numbers, how would you find number $n$?

48.

Which are your favourite topics in maths?

49.

How do you think science is portrayed in the media?

50.

What career are you interested in after your degree?

51.

Describe a complex number to a non-mathematician.

52.

Problem 44: Prove that the angle at the centre of a circle is twice that at the circumference.

53.

Problem 45: How many ways are there in which you can colour three equal portions of a disc?

54.

55.

56.

Problem 89: Explain what integration is.

57.

To integrate a function means to find a function the derivative of which is the given function.

58.

90. If $n$ is a perfect square and its second last digit is 7, what are the possibilities for the last digit of $n$ and can you show this will always be the case?

59.

Do you know where the multiplication sign came from (Oxford applicant, mathematics and statistics, The Student Room)? (注意概念性问题需要了解数学史)

60.

Asia Mathematics Cambridge

61.

Since I have signed the data protection form, I am not supposed to tell others exact interview questions. I can only give some general but useful information and tips about the interviews and questions. The feedback is mainly in Chinese because I used many Chinese books as reference when I prepared for interview and wrote this feedback, translation would make people confused. Sorry for breaking the English language policy.

62.

Cambridge interview Queens' College Asia 3476. Mathematics. 共两场面试，无 PS 相关，无 general question. 直接做题. 每场面试两个考官，每人问了一道题.

63.

第一题：关于 Regular Polyhedron.

64.

建议阅读：Wikipedia 中 Platonic Solid

65.

校图书馆里有本书叫 ``Algebra and geometry'' by A. Lan F. Beardon, 里面有一章关于 Regular Polyhedrons.

66.

第二题：与去年 ``Frank 3419 Maths Oxford Jesus'' 中的一道题十分相似

67.

第三题：关于 Complex Number.

68.

建议：1. 熟练掌握 FM 中 complex number 的知识点，尤其是熟用 $e^{i\theta} = \cos\theta + i\sin\theta$，并与 Argand diagram 结合.

69.

e.g. $z \cdot e^{i\theta}$ 表示逆时针旋转 $\theta^\circ$, $z_1 + z_2$ 可用平行四边形法则与向量加减结合.

70.

第四题：综合问题，涉及：perfect elastic collision；平面几何题（初中水平）；组合数学中的递推关系思想.

71.

其他：1．可能因为我 STEP 全过了，所以本应是必考的微积分、函数一道没问（这些也是 STEP 的重点），但它们仍较重要，建议有时间的话看一看数学分析的题目，推荐《数学分析中的典型问题与方法》裴礼文著，高等教育出版社，来不及就倒好 STEP 吧！

72.

2．如果虚组合数学（如排列组合），有时间推荐《组合数学引论》，许胤龙孙淑玲编著，中国科学技术大学出版社；来不及的话请刷好 S1 中的 Permutation \& Combination 也就够了。

73.

3．面试官颜值高也不要盯着人家看，不然就无法专心做题，面试就过不了，以后也见不到人家！

74.

本着“授人以鱼不如授人以渔”的指导思想和被 data protection form 限制，故没有直接写题，祝学弟学妹面试顺利！

75.

1 What is golden number? (PS 里提到过) Do you know anything about its apply? Why we said that the rectangle with golden ratio of its length and width is geometric perfect?

9. 几何 (Geometry) (100题)

If I have a triangle of fixed perimeter $P$ what will the maximum area be?

Does there exist a right-angled triangle of fixed perimeter $P$ of smallest area?

If you are faced with a corridor of width $m$ and another corridor of width $n$, which is perpendicular to the first, what is the maximum length of ladder you can carry through the corridors? You may model the ladder as a one-dimensional rod.

What do you know about triangles?

You are given a triangle with a fixed perimeter but you want to maximise the area. What shape will it be? Prove it.

Next you are given a quadrilateral with fixed perimeter and you want to maximise the area. What shape will it be? Prove it.

What the 2 sides of a rectangle ($a$ and $b$) would be to maximise the area if $a + b = 2c$ (where $c$ is a constant).

What is the area between two circles, radius one, that go through each other’s centres?

Prove Ptolemy’s Theorem.

10.

Prove that the angle at the centre of a circle is twice that at the circumference.

11.

Explain why the length sum of the diagonals of any quadrilateral is less than its perimeter.

12.

How many triangle centres do you know?

13.

What is the area of an $n$-sided regular polygon inscribed within a circle of radius $R$?

14.

Give a vector proof that for a triangle inscribed within a semicircle, the included angle is always $\pi/2$.

15.

Find the equation of the curve of the ladder as it falls (Consider $\theta$ fixed, not the mid-point). Sketch the solution of curve.

16.

(a) What’s the properties of isosceles triangles?

17.

(b) Prove in a circle, angle subtended by diameter is right-angle triangle.

18.

(c) Prove the converse is also true (that is: if $\triangle ABC$ is a right-angle triangle, then points $A$, $B$ and $C$ are on a same circle).

19.

(d) Draw a convex set and a nonconvex set (A convex set is a collection of points in which the line $AB$ connecting any two points $A,B$ in the set lies completely within the set.)

20.

(e) Now suppose we have a length of unit of 1. Consider the enclosed area, how to maximize the area? Guess what the shape should be?

21.

(f) Show that a non-convex set cannot achieve maximum area with given perimeter.

22.

(g) Consider a line segment, the line above this line segment has the same length as the one below it, but different areas. Show that with this perimeter, this shape cannot obtain the maximum area.

23.

(h) Prove that the shape in (e) should be a circle.

24.

The points $A(6,0)$ and $B(0,-4)$ are points on a triangle, the third point lies on the graph of $y = x^2$, find the coordinates of the third point which minimises the area of the triangle.

25.

Draw $y = \frac{e^\theta}{1+e^\theta} \sin\theta$ and $x = \frac{e^\theta}{1+e^\theta} \cos\theta$ in polar coordinate.

26.

A ladder is leaning against a vertical wall with its feet on the ground. The middle rung is painted yellow on the side, such that when we look at it side on, we can see it. What shape does this middle rung trace out as the ladder falls to the ground?

27.

What shape there would be if the cube was cut in half from diagonally opposite vertices?

28.

Rubik had a cube and held it by two diagonally opposite vertices and rotated it till it reached the same position, by how many degrees did it take a turn?

29.

Divide a cake, which is a cube, into 7 equal portions with same volume and surface area.

30.

How many faces are there on an icosahedron.

31.

An ant is at a vertex of a cube which has side length 1. If the ant can only walk across the faces of the cube, what is the shortest distance to the opposite vertex?

32.

What is the volume of the largest cube that fits entirely within a sphere of unity volume?

33.

A square piece of paper has a side length of 10 cm. What is the maximum volume that can be enclosed by the remaining portion by cutting away four squares at each of the four corners?

34.

How many distinct tessellations of the plane use only one regular polygon?

35.

Why are there only five platonic solids?

36.

Using Euler’s Polyhedron Formula $V - E + F = 2$ show that a platonic solid made of triangles must have 4, 8 or 20 faces.

37.

If $n$ points are distributed around the circumference of a circle and each point is joined to every other point by a chord of the circle (assuming that no three chords intersect at a point inside the circle), into how many regions is the circle divided?

38.

(a) How many regions can a circle be divided into by one line? two lines? three lines?

39.

(b) What is the maximum number of regions that a circle can be divided into by $n$ lines?

40.

If I have a triangle of fixed perimeter $P$, what will the maximum area be?

41.

If you are faced with a corridor of width $m$ and another corridor of width $n$, which is perpendicular to the first, what is the maximum length of ladder you can carry through the corridors?

42.

[You may model the ladder as a one-dimensional rod.]

43.

Prove Pick’s Theorem.

44.

Polygon with lattice vertices. Area $= I(p) + \frac{1}{2}B(p) - 1$

45.

$I(P)$: number of lattice points in interior $B(P)$: number of lattice points on boundary

46.

Assume there are $n$ rectangles that have a side ratio of $1:2$. How many $n$ can these rectangles consist in a square? (can be in different sizes)

47.

Prove Pythagoras’ Theorem.

48.

$$ (a + b)^2 = c^2 + 2 \cdot ab \Rightarrow a^2 + b^2 = c^2 $$

49.

Show that for a triangle $ABC$ which is not a right triangle, \[ \tan A + \tan B + \tan C = \tan A \cdot \tan B \cdot \tan C. \]

50.

Squares have unit length. Find area of $A$, $B$, $C$.

51.

52.

53.

How many diagonals can we draw in a pentagon? How about a polygon with $n$ sides?

54.

A graph of three straight lines and one curve is changed in a way that all straight lines are stretched into curves. The points of intersection are all unchanged. Find the curve.

55.

Find the angle between the diagonal of the cube to each face.

56.

What is the order of rotation for $AG$?

57.

58.

Note: The image contains a hand-drawn cube labeled with vertices $A, B, C, D, E, F, G, H$, but no explicit figure description is needed beyond the text content. The extracted text has some typos (e.g., "Phore" instead of "Prove", "intenor" instead of "interior"), which are corrected based on context and image.

59.

Prove that in an equilateral triangle, the sum of the distances from an interior point to each side is equal.

60.

There are $n$ points in a plane. Prove that there is a way to start from any point and connect all points and return to the original point without crossing edges.

61.

Is it possible to use 7 lines to cut a triangle into acute triangles only?

62.

Let $a$, $b$, $c$ be the sides of a triangle. Prove that \[ a^4 + b^4 + c^4 \leqslant 2(a^2b^2 + b^2c^2 + a^2c^2). \]

63.

An upside-down cone of height $h$ is filled halfway up. What proportion of the volume of the cone is empty?

64.

How many corners and sides does a square have in $n$ dimensions?

65.

What are the coordinates of the vertices of an octagon with sides of unit length and center at the origin?

66.

What is the maximum area of an isosceles triangle with two sides of unit length?

67.

Two identical cylinders of unit radius have axes intersecting at right angles, so that their intersection has four identical curved surfaces. Find the volume of this intersection without using calculus.

68.

(a) A hexagon has sides $a_1, a_2, a_3, \ldots, a_6$ with a circle inscribed inside that touches each side. Given that the hexagon is not necessarily regular, prove that the sum of alternate sides is equal, i.e.,

69.

\[ a_1 + a_3 + a_5 = a_2 + a_4 + a_6. \] (b) Now prove this for all even-sided polygons.

70.

(Homerton) A square piece of paper has side length $10$ cm. What is the maximum volume that can be enclosed by the remaining portion after cutting away four squares at each of the four corners?

71.

(Corpus Christi) Given three points inscribed on a circle, how should the points be placed to maximize the area of the triangle they form?

72.

will be located? How many will there be for this area?

73.

(Oriel) (a) What's the properties of isosceles triangles? (b) Prove in a circle, angle subtended by diameter is right-angle triangle. (c) Prove the converse is also true (that is: if $\triangle ABC$ is a right-angle triangle, then points $A$, $B$ and $C$ are on a same circle). (d) Draw a convex set and a nonconvex set (A convex set is a collection of points in which the line $AB$ connecting any two points $A,B$ in the set lies completely within the set.) (e) Now suppose we have a length of unit of 1. Consider the enclosed area, how to maximize the area? Guess what the shape should be? (f) Show that a non-convex set cannot achieve maximum area with given perimeter. (g) Consider a line segment, the line above this line segment has the same length as the one below it, but different areas. Show that with this perimeter, this shape cannot obtain the maximum area. (h) Prove that the shape in (e) should be a circle.

74.

Prove Pythagoras' theorem.

75.

How many different types, and numbers, of rotations on axis of symmetry can you have for a cube?

76.

Prove that $3 > \pi > 4$, considering the area of a circle compared to that of a square, and an $n$-sided shape placed inside the circle.

77.

Draw $y = e^x$ Then draw $y = kx$ Then draw a graph of the number of solutions of $x$ against $x$ for $e^x = kx$, Then find the value of $k$ where there is only 1 solution.

78.

Find the general formula for the sum of interior angles of a polygon.

79.

You are given a triangle with a fixed perimeter but you want to maximise the area. What shape will it be? Prove it. Next, you are given a quadrilateral with fixed perimeter and you want to maximise the area. What shape will it be? Prove it.

80.

A sector is cut from a circle and a cone is made from the remaining material by pulling the two freshly cut edges together. What should the angle of the sector be in order to maximise the volume of the cone?

81.

Problem 19: What shape would there be if a cube was cut in half from diagonally opposite vertices?

82.

(2) Cut the cube as shown. Find the relationship between the area of each side.

83.

(3) Differentiate $ y = \ln(\cos(x^4)) $. Sketch the graph.

84.

(4) Fractal. Find the area and circumference when $ n \to \infty $.

85.

Note: The fractal figure is not fully visible; it appears to be a recursive geometric construction such as the Koch snowflake or similar. Without full details, assume standard fractal behavior.

86.

(5) $n$ boys and girls sit in a circle. How many ways for different $n$.

87.

PS: Mary questions related to PS.

88.

interview: (20 minutes)\\ There are 20 points, and there are 100 segments. Is there always a triangle? Proof and explain.
There are 20 points, and there are 101 segments. Is there always a triangle? Proof and explain.

89.

Prove wrong: \\ ($S$ is area)\\ a hexagon of side length $a$, $b$, $c$, $d$, $e$, $f$, respectively.

90.

$S = abcdef$
$S = a^2 + \frac{1}{2}b^2 + \frac{1}{3}c^2 + \frac{1}{4}d^2 + \frac{1}{5}e^2 + \frac{1}{6}f^2$
$S = a^2 - a + b^2 - b + c^2 - c + d^2 - d + e^2 - e$

91.

1.a. Find the largest volume of the sphere inside the cube.

92.

b. find the largest volume of the cube inside the sphere.

93.

calculate the area of

94.

squares have unit length.

95.

3. If there is three strings, you randomly pair the 6 ends of them. What is the chance to get a complete circle?

96.

4. If there is a stick, you break it at point A then break it at point B. What is the chance that the three sections can make a triangle?

97.

The hardest interview question I was asked:

98.

The question is about polyhedron. More specifically, it’s about the relationship of vertice, edge and faces of the polyhedron.

99.

The professor asked me to give an example of polyhedron, which is easy. Then, he asked me to tell him the vertice, edge and faces of the example that I've given. Then he gave me a formula: \[ \text{Vertices} - \text{Edges} + \text{Faces} = 2 \] Then he told me which is the number of faces which is made $ F_n $ up of $ n $ edges in a polyhedron. Then, he asked me what is the smallest value of $ n $. I got it wrong at this, which is stupid.

100.

Then he asked me to write down the total number of faces in terms of $ F_n $. Then write down the total number of edges of a polyhedron in terms of $ F_n $, which I finished with some hints. Given $ F \geq V $, find the smallest value of $ F_3 $.

10. 复数 (Complex) (10题)

How many solutions does $z^3 = 1 + i$ have?

How many solutions does $z^\pi = 1 + i$ have?

What is the square root of $i$?

Let $z = 1 + 2i$. Show in an Argand diagram $z$, $3z$, $iz$, $|z|$ and $\sqrt{z}$.

$i = \cos\frac{\pi}{2} + i\sin\frac{\pi}{2}$

Let $(\cos\theta + i\sin\theta)^2 = i \Rightarrow \cos 2\theta + i\sin 2\theta = i \Rightarrow 2\theta = \frac{\pi}{2} + 2n\pi \Rightarrow \theta = \frac{\pi}{4} + n\pi \Rightarrow \theta = \frac{\pi}{4} \text{ or } \frac{\pi}{4} + \pi$

\[ \cos\theta + i\sin\theta = \cos\frac{\pi}{4} + i\sin\frac{\pi}{4} = \frac{1}{2}\sqrt{2} + \frac{1}{2}i\sqrt{2} \]

\[ \cos\theta + i\sin\theta = \cos\left(\frac{\pi}{4} + \pi\right) + i\sin\left(\frac{\pi}{4} + \pi\right) = -\frac{1}{2}\sqrt{2} - \frac{1}{2}i\sqrt{2} \]

$\sqrt{i} \to$ Calculate \& draw on Argand diagram

10.

$\sqrt[n]{i} \to$ guess the number of roots \& the shape of the diagram representing roots in terms of $n$. (without calculation)

11. 曲线绘制 (Sketch) (100题)

Sketch $ y = x \ln(x) $

Sketch $ y = \frac{\ln(x)}{x} $ and hence find all natural solutions of the equation $ a^b = b^a $.

Sketch $ y = x^x $ and $ y = x^{\frac{1}{x}} $

Sketch $ y = \frac{\sin(x)}{x} $ and $ y = \frac{\sin(x)}{x-1} $

Sketch $ y = \cos\left(\frac{1}{x}\right) $ and $ y = \sin\left(\frac{1}{x}\right) $

Sketch $ y = \frac{x + \sin(x)}{x - \sin(x)} $

Sketch $ y = \cos(x + |x|) $ for $ -2\pi < x < 2\pi $

Sketch $ y = |x^2 - 1| $, $ y = x^{\frac{1}{3}} $, $ y = x^{\frac{2}{3}} $ and comment on their derivative.

Sketch $ y = e^{-x^2} - e^{-3x^2} $

10.

By sketching appropriate graphs, find all solutions to the equation $ x - 1 = (e - 1)\ln(x) $. Hence sketch the graph with equation $ y = e^x - x^e $.

11.

Write $ \frac{3e^x - e^{-x}}{e^x + e^{-x}} $ in the form $ a + \frac{b}{e^{2x} + 1} $ and hence sketch $ y = \frac{3e^x - e^{-x}}{e^x + e^{-x}} $

12.

Sketch $ x^{2n} + y^{2n} = 1 $ for $ n = 2 $ and $ 4 $. Explain what happens to the graph as $ n \to \infty $.

13.

Sketch the curve $ |3x^2 + y^2 - 12| = |x^2 - y^2 + 4| $

14.

Sketch $ y = \sqrt{1 - x^2} + \sqrt{4 - x^2} $

15.

Draw $ y = A\left(1 - e^{-Bx}\right)^2 $

16.

Draw $ y = \cos(x^{\cos x}) $

17.

Draw: $ y = e^x $, $ y = e^{-x} $, $ y = \frac{e^x + e^{-x}}{2} $

18.

Sketch $ y = \frac{\ln x}{x} $ and find its maximum.

19.

Draw $ y = \frac{e^x - 1}{e^x + 1} $.

20.

By drawing a graph of $ f(x) = \frac{\ln x}{x} $, find out which is greater: $ e^x $ or $ x^e $?

21.

Draw the graph of $ y = x^5 - 3x^3 + 2x^2 $.

22.

Draw the graph of $ y = x^2 e^{-x} $.

23.

Draw $ y = x^2 $, $ y = x^4 $, and $ y = e^x $ on the same axes. As $ x \to \infty $, which is bigger?

24.

Sketch $ y = x(\ln x)^2 $.

25.

Sketch $ y = x \ln(x) $.

26.

Sketch $ y = \ln(\sin x) $, highlighting any maxima or minima.

27.

Sketch $ f(x) = (x - R(x))^2 $, where $ R(x) $ is $ x $ rounded up or down in the usual way. Then sketch $ g(x) = f(1/x) $.

28.

Sketch the graph of $ 1/x $, $ 1/x^2 $, $ 1/(1 + x^2) $.

29.

Sketch $ y = x^3 $ and $ y = x^5 $ on the same axis. Also for $ y = x^{103} $ and $ y = x^{105} $.

30.

Draw $ y = e^x $, $ y = \ln x $, $ y = x $. What does this show you?

31.

As $ x $ tends to infinity, what does $ \frac{\ln x}{x} $ tend to?

32.

Graph $ y = \max(1, x) $ for $ 0 \leq x \leq 3 $. What is the area under the graph?

33.

Graph $ y = x^{1/100} $ and $ y = x^{1/101} $.

34.

Draw the graph of $ x^2 \sin\left(\frac{1}{x}\right) $, $ 0 \leq x \leq \frac{2}{\pi} $.

35.

Draw the graph of $ f(x) = \frac{\ln x}{x} $, then find all integer solutions of $ a^b = b^a $, then $ a^{b^2} = b^{a^2} $? Take $ \ln $ on both sides and use the graph. Prove that no integer solutions exist for $ a^{b^2} = b^{a^2} $.

36.

Sketch $ y = x $ and $ y = x^3 $ on same axis. Sketch $ y = ax^3 $ (i) for $ 0 < a < 1 $, (ii) for $ a > 1 $. Sketch $ y = x - ax^3 $. Sketch $ ax^3 - x + 1 = 0 $ for $ a $ very very small.

37.

Draw $ e^{-x^2} $

38.

Draw $ \cos(x^2) $

39.

Sketch $ y = \sqrt{x^3 - x} $ and $ y^2 = x^3 - x $

40.

Sketch $ y = \frac{x^4 - 7x^2 + 12}{x^4 - 4x^2 + 4} $

41.

Sketch $ y = \frac{x^2 + 1}{x^2 - 1} $

42.

Sketch $ y = x^2 - x^4 $ and $ y^2 = x^2 - x^4 $ (consider the derivative at the origin carefully).

43.

Sketch $ y = x^3 - x^5 $, hence sketch $ y^2 = x^3 - x^5 $.

44.

Sketch $ \sin(1/x) $.

45.

Sketch $ y = x \sin x $

46.

Practise graphing any $ y = f(x) $ and then $ y = \frac{1}{f(x)} $.

47.

Graph $ y = \sin x, y = \sin^2 x, y = \sin^3 x, \cdots $, any thoughts on $ y = \sin^n x $.

48.

Find the number of integer solutions to the equation $ |x| + |y| \leq 100 $.

49.

Sketch $ 1 = |x| + |y| $, $ 1 = |x| - |y| $, and $ 1 = |y - x| $.

50.

What is the area of the region in the Cartesian plane whose points $ (x,y) $ satisfy $ |x| + |y| + |x + y| < 2 $?

51.

Sketch the curve $ (y^2 - 2)^2 + (x^2 - 2)^2 = 2 $.

52.

Sketch the following curve and identify all the symmetries, if any: $ y = \sin(x) + \cos(2x) $.

53.

Sketch the following curve and identify all of the symmetries, if any: $ y = \frac{1}{2}e^{-x^2 + 2x + 3} $.

54.

Sketch the following curve and identify all of the symmetries, if any: $ x^2 y^2 = \frac{3}{x + y} $.

55.

Sketch the following curve and identify all of the symmetries, if any: $ y = \sqrt{x^2 - x - 6} $.

56.

Draw the graph $ x^4 + y^4 = 1 $ (Hint: $ x^2 + y^2 = 1 $). Extension: $ x^{2n} + y^{2n} = 1 $.

57.

Plot $ (x - a)^2 + (y - b)^2 = 1 $. Plot $ (x - a)^4 + (y - b)^4 = 1 $.

58.

What does $ x^2 + y^2 + z^2 = 1 $ represent on a 3D graph? If a plane $ x + y + z = a $ intersects the above, what 2D shapes are made?

59.

Find the equation of the plane which makes each shape.

60.

Graph $ \frac{x^2}{4} + \frac{y^2}{9} = 1 $. What about $ \frac{x^2}{4} + \frac{y^2}{9} + \frac{z^2}{49} = 1 $ (in 3D)?

61.

Draw $ y = 4x(1 - x) $ and sketch $ y = kx + 1 $. For which values of $ k $ do these lines cross twice?

62.

(a) Sketch $ y = \ln x $, then discuss the graph, e.g., what happens as $ x $ gets big? What happens near $ x = 0 $?

63.

(b) Sketch $ y = mx $ for some other $ m $. What do you notice about the graph?

64.

How many intersections are there?

65.

How could you distinguish between the times when there are 0 intersections / 1 intersection / 2 intersections?

66.

(c) Looking for $ \frac{dy}{dx} = \frac{1}{x} = m $ and $ \ln(x) = mx $ and solve to find that if $ 0 < m < \frac{1}{e} $ there are two solutions, but if $ m > \frac{1}{e} $ there are no solutions and if $ m \leq 0 $ then there will only be one solution.

67.

Find roots of the equation $ mx = \sin x $, considering different values of $ m $.

68.

Draw $ y = e^x $, then draw $ y = kx $, then draw a graph of the numbers of solutions of $ x $ against $ x $ for $ e^x = kx $, and then find the value of $ k $ where there is only 1 solution.

69.

Sketch $ x^2 - n y^2 = 0 $ where $ n $ is a natural number.

70.

Find all natural solution pairs $ (x, y) $ in the case $ n = 9 $.

71.

Find all natural solution pairs $ (x, y) $ in the case $ n = 10 $.

72.

Sketch $ y = x \ln(x) $.

73.

Sketch $ y = \frac{\ln(x)}{x} $ and hence find all natural solutions of the equation $ a^b = b^a $.

74.

Sketch $ y = (x)^x $ and $ y = (x)^{\frac{1}{x}} $

75.

Sketch $ y = |x^2 - 1| $ and comment on the derivative.

76.

Sketch $ y = x^{\frac{1}{3}} $ and comment on the derivative.

77.

Sketch $ y = x^{\frac{2}{3}} $ and comment on the derivative.

78.

Sketch $ y = x^2 - x^4 $ and $ y^2 = x^2 - x^4 $ (consider the derivative at the origin carefully)

79.

Write $ \frac{3e^x - e^{-x}}{e^x + e^{-x}} $ in the form $ a + \frac{b}{e^{2x} + 1} $ and hence sketch $ y = \frac{3e^x - e^{-x}}{e^x + e^{-x}} $.

80.

Sketch $ x^{2n} + y^{2n} = 1 $ for $ n = 2 $ and $ 4 $.

81.

Explain what happens to the graph as $ n \to \infty $.

82.

Sketch the curve $ |3x^2 + y^2 - 12| = |x^2 - y^2 + 4| $.

83.

Sketch

i) $ 1 = |x| + |y| $
ii) $ 1 = |x| - |y| $
iii) $ 1 = |y - x| $

84.

By considering the graph of the function $ f(x) = x^{-s} $, show that \[ \frac{1}{s-1} < 1 + 2^{-s} + 3^{-s} + \cdots < \frac{s}{s-1} \] whenever $ s > 1 $.

85.

Given $ x = a \cos^3 t $, $ y = a \sin^3 t $, $ a > 0 $, $ 0 \leq t \leq 2\pi $, find an equation containing $ x, y, a $ only and plot it.

86.

Sketch $ y = \cos^2 x \sin x $ \quad $ (0 \leq x \leq 2\pi) $.

87.

How many solutions does $ kx = e^x $ have for different values of $ k $?

88.

Compare $ e^\pi $ and $ \pi^e $.

89.

Sketch $ y = e^x - x^e $

90.

Sketch $ y = \tan x \ln(x - 1) $

91.

Sketch $ x = \sin t $, $ y = \sin t \cos t $

92.

Sketch $ \frac{1}{x^2 - 1} $

93.

Is $ \displaystyle \int_1^\infty \sin\left( \frac{1}{x^2 - 1} \right) dx $ finite or infinite?

94.

Plot $ \ln(\ln x) $

95.

Plot $ \ln(\sin x) $

96.

Sketch $ x^y = y^x $, $ x, y > 0 $

97.

\[ \frac{\ln y}{y} = \frac{\ln x}{x} \]

98.

Sketch the graph of $\frac{1}{x^2+1}$ and prove $\pi > 3$.

99.

Sketch $\frac{dy}{dx} = (2 - y)(y - 1)$.

100.

Plot $x^2 - y^2 - z^2 = 1$.