# Mathematics

In the first two rounds of the Mathematical Olympiad, three areas of mathematics are covered: number theory, combinatorics and geometry. In the final round, algebra will be added. Here you will find a short description for each of the four topics as well as two tricky example problems each.

## Number Theory

Number theory deals with the **integers** $\dots, -2, -1, 0, 1, 2,\dots$ and their innumerable exciting properties. We are mainly interested in the concepts of **divisibility** and, in particular, **prime** **numbers**: We say that the integer $a$ is divisible by the integer $b$ if we can divided $a$ by $b$ without a remainder. Thus, a prime number is a natural number that is only divisible by exactly two different natural numbers, by $1$ and by itself.

## Example Problems

**Problem 1:** Let $n$ be an odd integer. Prove that the number $n^3 – n$ is definitely divisible by $24$.

**Problem 2: **Let $n$ be a positive integer. Prove that at most two of the three numbers $n + 5$, $n + 7$ and $n + 9$ are prime numbers.

## Combinatorics

Combinatorics is all about investigating, counting, arranging or constructing discrete mathematical structures. The main focus is on the concepts of **bijection**, **pigeonhole principle **and **induction**. Typical problems often start with "*How many possibilities are there to* ..." or "*Is it possible that* ...".

## Example Problems

**Problem 1: **Prove that among each group of $6$ people there are always either $3$ people who are all friends $3$ people who are all not friends

**Problem 2: ** Two opposite corner squares are removed from an $8 \times 8$ chess board. Is it possible to cover all remaining squares with $31$ dominos, assuming one domino covers two adjacent squares?

## Geometry

In geometry we deal exclusively with **points**, **lines** and **circles**. A typical problem describes a general geometric configuration and the task is to prove a certain property which holds for each possible such configuration. The main focus is on the concepts of **length ratios**, **angles** and **symmetry**. Although careful construction with ruler and compass is not directly part of the competition, it can often be of great advantage.

## Example Problems

**Problem 1: ** Prove that in any convex polygon the sum of all interior angles is equal to $180\cdot (n-2)$ degrees, where $n$ is the number of vertices of the polygon.

**Problem 2: **Prove that in every triangle all three altitudes intersect in a single point.

## Algebra

Algebra is not part of the Mathematical Olympiad until the final round and is divided further into four areas: The **inequalities** are about proving that certain algebraic terms are always smaller or larger than others. The **functional****equations** are about finding all functions that fulfill certain conditions. There are also tasks concerning **sequences**, which typically involve proving certain properties of infinite sequences of numbers. In the selection round the theory of **polynomials** is added.

## Example Problems

**Problem 1:** Prove that for arbitrary real numbers $a$, $b$ and $c$ the inequality

$a^2 + b^2 + c^2 \geq ab + bc + ca$ always holds.

**Problem 2: **Find all functions $f$, defined from the rational numbers to the rational numbers, for which $f(x+y) = f(x) + f(y)$ holds for all rational numbers $x$, $y$.

## Ressources

If you want to know what you can expect in terms of theory at the Mathematical Olympiad, have a look at our ** scripts**. More resources and useful tips for preparation can be found under

**training**. If you want to solve more Olympiad style tasks to prepare for the competition or just for fun, check out the

**past exams**. Be aware, however, that the format and difficulty of the different rounds has changed in recent years.