Moldova, Third Team Selection Test for IMO and BMO 2008, March 30.

Posted on April 1, 2008 by aimingbeyond

Moldova, Third Team Selection Test for IMO-BMO 2008, March 30

Determine a subset $A\subset \mathbb N^*$ having exactly $5$ distinct elements, whose sum of squares equals their product.

Problem 2

Let $p$ be a prime number and $k,n$ positive integers so that $\gcd(p,n)=1$ . Prove that $\displaystyle\binom{n\cdot p^k}{p^k}$ and $p$ are coprime.

Problem 3

In triangle $ABC$ the bisector of $\angle{ACB}$ intersects $AB$ at $D$ . Consider an arbitrary circle $O$ passing through $C$ and $D$ and not tangent to $BC$ or $CA$ . Let $O\cap BC=\{M\}$ and $O\cap CA=\{N\}$ .

Prove that there is a circle $S$ so that $DM$ and $DN$ are tangent to $S$ in $M$ and $N$ respectively.
Circle $S$ intersects $BC$ and $CA$ in $P$ and $Q$ respectively. Prove that the lengths of $MP$ and $NQ$ do not depend on the choice of circle $O$ .

Problem 4

A non-empty set $S$ of positive integers is said to be good if there is a coloring with $2008$ colors of all positive integers so that no number in $S$ is the sum of two different positive integers (not necessarily in $S$ ) of the same color. Find the largest value $t$ can take so that the set $S=\{a+1,a+2,\ldots,a+t\}$ is good, for any positive integer $a$ .