ELMO 2003

Let $ABCDEF$ be a convex equilateral hexagon with sides of length $1$ . Let $R_1$ be the area of the region contained within both $ACE$ and $BDF$ , and let $R_2$ be the area of the region within the hexagon outside both triangles. Prove that: $$\min \{ [ACE], [BDF] \} + R_2 - R_1 \le \frac{3\sqrt{3}}{4}.$$

In a set of $30$ MOPpers, prove that some two MOPpers have an even number of common friends.

Let $k$ be a positive integer for which the equation $$2ab+2bc+2ca-a^2-b^2-c^2 = k$$ has some solution in positive integers $a,b,c$ . Prove that the equation has a solution for which $a$ , $b$ and $c$ are the sides of a possibly degenerate triangle.

Let $x,y,z \ge 1$ be real numbers such that $$\frac{1}{x^2-1} + \frac{1}{y^2-1} + \frac{1}{z^2-1} = 1.$$ Prove that $$\frac{1}{x+1} + \frac{1}{y+1} + \frac{1}{z+1} \le 1.$$