Maths Oxbridge Interview Questions

Twenty genuine-difficulty maths questions to help you prepare for your interview, written and illustrated by our Oxbridge-graduate tutors. Both Oxford and Cambridge now require the TMUA for Mathematics, so if you are preparing for interview you will be sitting it too. Our TMUA past papers and resources has every paper from 2016 to 2023 with mark schemes.

Cambridge mathematics offers also name STEP. Our STEP past papers and resources has every paper from 2014 to 2025.

Question 1

Show that \( n^5 - n \) is divisible by 30 for every integer \( n\).

Show a hint

Treat divisibility by 2, 3 and 5 separately. Factor \( n^5 - n = (n-1)\,n\,(n+1)(n^2+1) \): three consecutive integers force factors of 2 and 3. For the factor of 5, consider the possible remainders of \(n\) when divided by 5 (or recall a result of Fermat). Show the expression is divisible by each of 2, 3 and 5.

Question 2

Prove \( \log_2 3 \) is irrational; you may assume \( \sqrt{2} \) is irrational.

Show a hint

Suppose \( \log_2 3 = p/q \) for positive integers \(p, q\), and rewrite that as an equation relating a power of 2 to a power of 3. Then ask what it says about one side being even while the other is odd.

Question 3

How many zeros does \( 100! \) end in?

Show a hint

A trailing zero comes from a factor of 10, that is, a pair of 2 and 5. Since factors of 2 are far more common than factors of 5, just count how many factors of 5 appear among the numbers up to 100, remembering that 25, 50, 75 and 100 each contribute more than one.

Question 4

Prove there are infinitely many primes of the form \( 4n + 3 \)

Show a hint

Argue by contradiction, in the spirit of Euclid. Assume only finitely many primes of the form \(4n+3\) exist, and build a new number from them that is itself of the form \(4n+3\). Show it must have a prime factor of that same form which is not on your list.

Question 5
An 8×8 chessboard has two diagonally opposite corners removed. Can it be tiled by 31 dominoes?

Show a hint

Colour the board in the usual two colours and ask what colour the two removed corners are. Then think about what one domino always covers, and compare how many squares of each colour are left.

Question 6
What is the greatest number of pieces a pizza can be cut into with n straight cuts?

Show a hint

Build it up one cut at a time. When you add the \(k\)-th cut, how many of the existing cuts can it cross, and how many new pieces does each crossing create? Turn that into a recurrence for the number of pieces and sum it.

Question 7
You paint each face of a cube a different colour from a set of six. How many distinct cubes are there, counting rotations as the same?

Show a hint

First count the colourings as if the cube were fixed in space, ignoring rotation. Then account for the fact that each genuinely distinct cube can be turned to look like several of those fixed arrangements. How many rotations does a cube have?

Question 8
n people sit randomly at a round table. What is the probability two particular people sit next to each other?

Show a hint

Fix one of the two people in a seat, then ask where the second person can sit. How many seats are available to them, and how many of those are immediately next to the first person?

Question 9

Evaluate \[ \sqrt{1+\sqrt{1+\sqrt{1+\cdots}}} \]

Show a hint

Notice that the whole expression reappears inside itself. Set it equal to \(x\), which gives \( x = \sqrt{1+x} \), then form an equation in \(x\) and choose the root consistent with a positive square root.

Question 10

Find x such that the infinite power tower \[ x^{x^{x^{\cdots}}} = 2 \] Follow up: Why does asking for the tower to equal 4 give the same x, and what does that tell you about convergence?

Show a hint

The exponent sitting on the base is the entire tower again, so replace it with the value the tower is said to equal. That collapses the problem to a simple equation for \(x\). For the follow-up, repeat the step assuming the tower equals 4, compare the two, and then think carefully about whether a tower can really converge to a value that large.

Ten more to go

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Question 11

Show that \( x + \dfrac{1}{x} \ge 2 \) for \( x > 0 \), then that \[ (a + b + c)\left( \frac{1}{a} + \frac{1}{b} + \frac{1}{c} \right) \ge 9 \] for positive \( a, b, c \).

Show a hint

For the first inequality, consider \( \left(\sqrt{x} - \tfrac{1}{\sqrt{x}}\right)^2 \), which cannot be negative. For the second, expand the product into nine terms and group them into pairs of the form \( \tfrac{a}{b} + \tfrac{b}{a} \), then apply the first result to each pair.

Question 12

A function satisfies \( f(x + y) = f(x)\,f(y) \) for all real \( x, y \), and isn't identically zero. What can \( f \) be?

Show a hint

Find \( f(0) \) first by setting \( x = y = 0 \). Then show \( f(x) \) is never negative by writing \(x\) as a sum of two halves, and ask which familiar family of functions turns addition of inputs into multiplication of outputs.

Question 13

Evaluate \[ \frac{1}{1\cdot 2} + \frac{1}{2\cdot 3} + \cdots + \frac{1}{n(n+1)}. \]

Show a hint

Split the general term using partial fractions: \( \tfrac{1}{k(k+1)} = \tfrac{1}{k} - \tfrac{1}{k+1} \). Write the sum out with this and watch which terms cancel.

Question 14

Sketch \( y = x^{x} \) for \( x > 0 \).

Show a hint

Write \( y = e^{x \ln x} \) and differentiate with the chain rule to locate the turning point. Then examine the behaviour as \(x\) approaches 0 from the right and as \(x\) grows large, and join the pieces into a single curve.

Question 15

Sketch \( y = \dfrac{\sin x}{x} \)

Show a hint

It is an even function, so work out \(x > 0\) and reflect. Find the value it approaches as \(x \to 0\), locate the zeros, and note that the size of the oscillations is controlled by \( \tfrac{1}{x} \).

Question 16

Sketch the curve \( y^{2} = x^{3} - x \).

Show a hint

Since \( y^2 \) cannot be negative, first find where \( x^3 - x \) is non-negative; that fixes which \(x\)-values are allowed. The curve is symmetric about the \(x\)-axis. Look at where \( y = 0 \), and how the allowed regions behave.

Question 17

Evaluate \[ \int_{0}^{\infty} x\,e^{-x^{2}}\,dx \qquad\text{and}\qquad \int_{0}^{\infty} x^{3}\,e^{-x^{2}}\,dx. \]

Show a hint

For the first integral, the substitution \( u = x^2 \) turns it into a standard exponential integral. For the second, the same substitution leaves an extra factor of \( x^2 = u \), which you can then finish by parts.

Question 18

How many real solutions does \( 2^{x} = x^{2} \) have?

Show a hint

Sketch \( y = 2^x \) and \( y = x^2 \) on the same axes and count where they meet. Two intersections are easy to find by trying small integers; the third needs you to look at negative \(x\), where one curve stays positive while the other climbs.

Question 19

What is \( i^{\,i} \)?

Show a hint

Write \(i\) in the form \( e^{i\theta} \) for a suitable angle, then raise it to the power \(i\). Watch what happens when \(i\) multiplies \(i\) in the exponent, and remember there is more than one valid choice of angle.

Question 20

Evaluate \[ \int_{0}^{\pi/2} \frac{dx}{1 + \tan x}. \]

Show a hint

Use the substitution that replaces \(x\) with \( \tfrac{\pi}{2} - x \) across this interval, and add the new integral to the original. The two combine into something very simple, which lets you read off the value.

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