2008 Polish Mathematical Olympiad Third Round
Contents
[hide]Day 1
Problem 1
In fields of table are written numbers
, where the numbers
are in the first row (from left side to right), numbers
in the second, etc.
In that table
fields are chosen, from which no two lie in the same row or column. Let
be the chosen number in row number
. Prove that
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.
Day 2
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
The areas of all cross sections of the parallelepiped with planes going through the middles of three of its edges, of which none two are parallel and have no common points, are equal. Prove that
is a cuboid.
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
.