JBMO 1999

Let $a,b,c,x,y$ be five real numbers such that $a^3 + ax + y = 0$ , $b^3 + bx + y = 0$ and $c^3 + cx + y = 0$ . If $a,b,c$ are all distinct numbers prove that their sum is zero.

Ciprus

For each nonnegative integer $n$ we define $A_n = 2^{3n}+3^{6n+2}+5^{6n+2}$ . Find the greatest common divisor of the numbers $A_0,A_1,\ldots, A_{1999}$ .

Romania

Let $S$ be a square with the side length 20 and let $M$ be the set of points formed with the vertices of $S$ and another 1999 points lying inside $S$ . Prove that there exists a triangle with vertices in $M$ and with area at most equal with $\frac 1{10}$ .

Yugoslavia

Let $ABC$ be a triangle with $AB=AC$ . Also, let $D\in[BC]$ be a point such that $BC>BD>DC>0$ , and let $\mathcal{C}_1,\mathcal{C}_2$ be the circumcircles of the triangles $ABD$ and $ADC$ respectively. Let $BB'$ and $CC'$ be diameters in the two circles, and let $M$ be the midpoint of $B'C'$ . Prove that the area of the triangle $MBC$ is constant (i.e. it does not depend on the choice of the point $D$ ).

Greece