The International Mathematics Olympiad (IMO, also known as the International Mathematical Olympiad) is an annual mathematics competition for high school students [IMO Article in Wikipedia]. It is one - in fact, the oldest - of theInternational Science Olympiads. The first IMO was held in Romania in 1959.
For this olympiad, the problems come from various areas of mathematics, such as are included in math curricula at secondary schools. Finding the solutions of these problems, however, requires exceptional mathematical ability and excellent mathematical knowledge on the part of the contestants.
The selection process for the IMO varies greatly by country. In some countries, especially those in east Asia, the selection process involves several difficult tests of a difficulty comparable to the IMO itself.
The participants are ranked based on their individual scores. Medals are awarded to the highest ranked participants, such that slightly less than half of them receive a medal. Subsequently the cutoffs (minimum scores required to receive a gold, silver or bronze medal respectively) are chosen such that the ratio of gold to silver to bronze medals awarded approximates 1:2:3. Participants who do not win a medal but who score seven points on at least one problem receive an honorable mention
Number Theory, including
Fundamental Theorems on Arithmetic
Linear and quadratic Diophantine equations, including Pell's equation
Arithmetic of residues modulo n, Fermat's and Euler's theorems
Algebra, including
Fundamental Theorems on Algebra, e.g. inequalities, factorization of a polynomial into a product of irreducible polynomials
Symmetric polynomials of several variables, Vieta's theorem
Combinatorics, including
Graph theory
Geometry, including
Properties of the orthocentre, Euler's line, nine-point-circle, Simson line, Ptolemy's inequality, Ceva and Menelaus etc.
https://en.wikipedia.org/wiki/International_Mathematical_Olympiad
http://olympiads.win.tue.nl/imo/
http://www.sofworld.org/