OMTC Prepare – the invariants

Syllabus and Format

You can download the format and syllabus of the OMTC here (last updated: 28th February, 2025).

The Oxford Mathematics Team Challenge is a competition between teams composed of four students, which consists of four rounds. One round is individual, and the other three are team rounds: teamwork is key!

The syllabus consists of most of the pure AS Level Mathematics, with the omission of all calculus, and with the addition of various topics from the GCSE and A-Level Mathematics curricula, among others.

Problem of the Day

Each weekday until the OMTC, we’ll share a problem of the day so that you can start the fun early! Solutions are available when you click the dropdown button.

Monday, 3rd March

A bug walks all day and sleeps all night. On the first day, it starts at point

O

, faces east, and walks a distance of

5

units due east. Each night the bug rotates

60

counterclockwise. Each day it walks in this new direction half as far as it walked the previous day. The bug gets arbitrarily close to the point

P

. Then

OP^2=\frac mn

, where

m

and

n

are relatively prime positive integers. Find

m+n

. (from AIME, 2020)

The bug goes forward and backward in three directions: straight east, northeast, and northwest. It travels $5-\tfrac58+\tfrac5{64}-\cdots=\tfrac{40}9$ units east. Thus, it goes $\tfrac{40/9}2=\tfrac{20}9$ units northeast and $\frac{10}9$ units northwest. Because of the $30-60-90$ right triangles in its path, the bug travels a total of $\frac{40}9+\frac{10}9-\frac59=\frac{45}9=5$ units due east and a total of $\tfrac{10\sqrt3}9+\tfrac{5\sqrt3}9=\tfrac{15\sqrt3}9=\tfrac{5\sqrt3}3$ units due north. Now, $OP^2=5^2+\tfrac{5^2}3=\tfrac{100}3$ by the Pythagorean Theorem, and the answer is $103$ .

Tuesday, 4th March

The amount

2.5

is split into two nonnegative real numbers uniformly at random, for instance, into

2.143

and

.357

, or into

\sqrt3

and

2.5-\sqrt3

. Then each number is rounded to its nearest integer, for instance,

2

and

0

in the first case above,

2

and

1

in the second. What is the probability that the two integers sum to

3

? (from AHSME, 1987)

Let $x$ be the smaller of the two numbers and $y$ the greater. Then either:

$x=0$ and $y=2.5$ , hence we round to $0$ and $3$ and get $3$
$0<x<0.5$ so that $2<y<2.5$ , hence we round to $0$ and $2$ and get $2$
$0.5\le x\le 1$ and $1.5\le y\le 2$ , hence we round to $1$ and $2$ and get $3$
$1<x\le1.25$ and $1.25\le y<1.5$ , hence we round to $1$ and $1$ and get $2$

Putting the potential values for $x$ on a line, we see that this line has length $1.25$ . The parts where the sum is $3$ have total length $0+0.5=0.5$ , so that we have a $\tfrac{0.5}{1.25}=\tfrac25$ chance of getting $3$ .

Wednesday, 5th March

A point whose coordinates are both integers is called a lattice point. How many lattice points lie on the hyperbola

x^2-y^2=2000^2

? (from AIME II, 2000)

(x-y)(x+y)=2000^2=2^8\cdot 5^6

Note that $(x-y)$ and $(x+y)$ are either both even or both odd. Since they multiply to an even number, they must both be even. We first give a factor of $2$ to both $(x-y)$ and $(x+y)$ . We have $2^6\cdot 5^6$ left. Since there are $7\cdot 7=49$ factors of $2^6\cdot5^6$ , and since both $x$ and $y$ can be negative, this gives us $49\cdot2=98$ lattice points.

Frequently Asked Questions

What kinds of maths can I expect to come up? We can’t tell you! A lot of the maths will be familiar and just use familiar tools in creative ways. Some of the maths will be new, but will be accessible to you. Make sure you’re comfortable with using the mathematicals tools in the syllabus.