Moldova, First Team Selection Test for IMO and BMO, 2008, March 3.

Posted on March 3, 2008 by aimingbeyond

Here goes the pdf file. Time allowed – 4 hours 30 mintues. Each problem is worth 7 points.

Moldova, First Team Selection Test for IMO-BMO 2008, March 3

Let $p$ be a prime number. Solve in $\mathbb{N}_0\times\mathbb{N}_0$ the equation $x^3+y^3-3xy=p-1$ .

We say the set $\{1,2,\ldots,3k\}$ has property $D$ if it can be split into disjoint triples, so that in each such triple, one number is the sum of the other two. Prove that

The set $\{1,2,\ldots,3324\}$ has property $D$ .
The set $\{1,2,\ldots,3309\}$ hasn’t property $D$ .

Problem 3

Let $\Gamma(I,r)$ and $\Gamma(O,R)$ be the incircle and circumcircle, respectively, of triangle $ABC$ . Consider all triangles $A_iB_iC_i$ which are simultaneously inscribed in $\Gamma(O,R)$ and circumscribed to $\Gamma(I,r)$ . Prove that the centroids of the triangles $A_iB_iC_i$ lie on a circle.

Problem 4

A non-zero polynomial $S\in\mathbb{R}[X,Y]$ is called homogeneous of degre $d$ if there is a positive integer $d$ so that $S(\lambda x,\lambda y)=\lambda^dS(x,y)$ for any $\lambda\in\mathbb{R}$ . Let $P,Q\in\mathbb{R}[X,Y]$ so that $Q$ is homogeneous and $P$ divides $Q$ (that is $P|Q$ ). Prove that $P$ is homogeneous too.