JBMO 1997

Show that given any 9 points inside a square of side 1 we can always find 3 which form a triangle with area less than $\frac 18$ .

Bulgaria

Let $\frac{x^2+y^2}{x^2-y^2} + \frac{x^2-y^2}{x^2+y^2} = k$ . Compute the following expression in terms of $k$ : $$E(x,y) = \frac{x^8 + y^8}{x^8-y^8} - \frac{ x^8-y^8}{x^8+y^8}.$$ Ciprus

Let $ABC$ be a triangle and let $I$ be the incenter. Let $N$ , $M$ be the midpoints of the sides $AB$ and $CA$ respectively. The lines $BI$ and $CI$ meet $MN$ at $K$ and $L$ respectively. Prove that $AI+BI+CI>BC+KL$ .

Greece

Determine the triangle with sides $a,b,c$ and circumradius $R$ for which $R(b+c) = a\sqrt{bc}$ .

Romania