2007 BMO Problems

Problems of the 2007 Balkan Mathematical Olympiad.

Problem 1

Let $\displaystyle ABCD$ be a convex quadrilateral with $\displaystyle AB=BC=CD$, $\displaystyle AC$ not equal to $\displaystyle BD$, and let $\displaystyle E$ be the intersection point of its diagonals. Prove that $\displaystyle AE=DE$ if and only if $\angle BAD+\angle ADC = 120^{\circ}$.


Problem 2

Find all functions $\displaystyle f : \mathbb{R} \mapsto \mathbb{R}$ such that

$\displaystyle f(f(x) + y) = f(f(x) - y) + 4f(x)y$, for any $x,y \in \mathbb{R}$.


Problem 3

Find all positive integers $\displaystyle n$ such that there exists a permutation $\displaystyle \sigma$ on the set $\{ 1, \ldots, n \}$ for which

$\sqrt{\sigma(1) + \sqrt{\sigma(2) + \sqrt{ \cdots  + \sqrt{\sigma(n)}}}}$

is a rational number.

Note: A permutation of the set $\{ 1, 2, \ldots, n \}$ is a one-to-one function of this set to itself.


Problem 4

For a given positive integer $\displaystyle n >2$, let $\displaystyle C_{1},C_{2},C_{3}$ be the boundaries of three convex $\displaystyle n$-gons in the plane such that $C_{1}\cap C_{2}$, $C_{2}\cap C_{3}$, $C_{3}\cap C_{1}$ are finite. Find the maximum number of points in the set $C_{1}\cap C_{2}\cap C_{3}$.