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 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.
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 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 ...".
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?
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.
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 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 functionalequations 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.
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 $x$, $y$.
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.