BG-MO 2003

Let $x_1, x_2 \ldots , x_5$ be real numbers. Find the least positive integer $n$ with the following property: if some $n$ distinct sums of the form $x_p+x_q+x_r$ (with $1\le p<q<r\le 5$ ) are equal to $0$ , then $x_1=x_2=\cdots=x_5=0$ .

Let $H$ be an arbitrary point on the altitude $CP$ of the acute triangle $ABC$ . The lines $AH$ and $BH$ intersect $BC$ and $AC$ in $M$ and $N$ , respectively. [list] (a) Prove that $\angle NPC =\angle MPC$ . (b) Let $O$ be the common point of $MN$ and $CP$ . An arbitrary line through $O$ meets the sides of quadrilateral $CNHM$ in $D$ and $E$ . Prove that $\angle EPC =\angle DPC$ . [/list]

Given the sequence $\{y_n\}_{n=1}^{\infty}$ defined by $y_1=y_2=1$ and $$y_{n+2} = (4k-5)y_{n+1}-y_n+4-2k, n\ge1$$ find all integers $k$ such that every term of the sequence is a perfect square.

A set $A$ of positive integers is called uniform if, after any of its elements removed, the remaining ones can be partitioned into two subsets with equal sum of their elements. Find the least positive integer $n>1$ such that there exist a uniform set $A$ with $n$ elements.

Let $a,b,c$ be rational numbers such that $a+b+c$ and $a^2+b^2+c^2$ are equal integers. Prove that the number $abc$ can be written as the ratio of a perfect cube and a perfect square which are relatively prime.

Determine all polynomials $P(x)$ with integer coefficients such that, for any positive integer $n$ , the equation $P(x)=2^n$ has an integer root.