# 1999 JBMO Problems

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## Problem 1

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.

## Problem 2

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 $A_0,A_1,\ldots, A_{1999}$.

## Problem 3

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 $\frac 1{10}$.

## Problem 4

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 $\frac 1{10}$.

## See Also

 1999 JBMO (Problems • Resources) Preceded by1998 JBMO Problems Followed by2000 JBMO Problems 1 • 2 • 3 • 4 All JBMO Problems and Solutions

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