OMCC 2016

Find all positive integers $n$ that have 4 digits, all of them perfect squares, and such that $n$ is divisible by 2, 3, 5 and 7.

Let $ABC$ be an acute-angled triangle, $\Gamma$ its circumcircle and $M$ the midpoint of $BC$ . Let $N$ be a point in the arc $BC$ of $\Gamma$ not containing $A$ such that $\angle NAC= \angle BAM$ . Let $R$ be the midpoint of $AM$ , $S$ the midpoint of $AN$ and $T$ the foot of the altitude through $A$ . Prove that $R$ , $S$ and $T$ are collinear.

The polynomial $Q(x)=x^3-21x+35$ has three different real roots. Find real numbers $a$ and $b$ such that the polynomial $x^2+ax+b$ cyclically permutes the roots of $Q$ , that is, if $r$ , $s$ and $t$ are the roots of $Q$ (in some order) then $P(r)=s$ , $P(s)=t$ and $P(t)=r$ .

The number "3" is written on a board. Ana and Bernardo take turns, starting with Ana, to play the following game. If the number written on the board is $n$ , the player in his/her turn must replace it by an integer $m$ coprime with $n$ and such that $n<m<n^2$ . The first player that reaches a number greater or equal than 2016 loses. Determine which of the players has a winning strategy and describe it.

We say a number is irie if it can be written in the form $1+\dfrac{1}{k}$ for some positive integer $k$ . Prove that every integer $n \geq 2$ can be written as the product of $r$ distinct irie numbers for every integer $r \geq n-1$ .

Let $\triangle ABC$ be triangle with incenter $I$ and circumcircle $\Gamma$ . Let $M=BI\cap \Gamma$ and $N=CI\cap \Gamma$ , the line parallel to $MN$ through $I$ cuts $AB$ , $AC$ in $P$ and $Q$ . Prove that the circumradius of $\odot (BNP)$ and $\odot (CMQ)$ are equal.