2008 Polish Mathematical Olympiad Third Round
Contents
[hide]Day 1
Problem 1
Problem 2
Function of three real variables satisfies for all real numbers
the equality
Prove that for all real numbers
the equality
is satisfied.
Problem 3
In a convex pentagon , where
, the equations
hold. Prove that
is a parallelogram.
Day2
Problem 4
Every point with integer coordinates on a plane is painted either black or white. Prove that among the set of all painted points there exists an infinite subset which has a center of symmetry and has all the points of the same colour.
Problem 5
Problem 6
Let be the set of all positive integers which can be expressed in the form
for some coprime integers
and
. Let
be a prime number with rest 3 when divided by 4. Prove that if some positive multiple of
belongs to
, then the number
also belongs to
.