RIOPLATENSE 2017

Let $a$ be a fixed positive integer. Find the largest integer $b$ such that $(x+a)(x+b)=x+a+b$ , for some integer $x$ .

One have $n$ distinct circles(with the same radius) such that for any $k+1$ circles there are (at least) two circles that intersects in two points. Show that for each line $l$ one can make $k$ lines, each one parallel with $l$ , such that each circle has (at least) one point of intersection with some of these lines.

Show that there are infinitely many pairs of positive integers $(m,n)$ , with $m<n$ , such that $m$ divides $n^{2016}+n^{2015}+\dots+n^2+n+1$ and $n$ divides $m^{2016}+m^{2015} +\dots+m^2+m+1$ .

Is there a number $n$ such that one can write $n$ as the sum of $2017$ perfect squares and (with at least) $2017$ distinct ways?

Let $ABC$ be a triangle and $I$ is your incenter, let $P$ be a point in $AC$ such that $PI$ is perpendicular to $AC$ , and let $D$ be the reflection of $B$ wrt circumcenter of $\triangle ABC$ . The line $DI$ intersects again the circumcircle of $\triangle ABC$ in the point $Q$ . Prove that $QP$ is the angle bisector of the angle $\angle AQC$ .

For each fixed positiver integer $n$ , $n\geq 4$ and $P$ an integer, let $(P)_n \in [1, n]$ be the smallest positive residue of $P$ modulo $n$ . Two sequences $a_1, a_2, \dots, a_k$ and $b_1, b_2, \dots, b_k$ with the terms in $[1, n]$ are defined as equivalent, if there is $t$ positive integer, gcd $(t,n)=1$ , such that the sequence $(ta_1)_n, \dots, (ta_k)_n$ is a permutation of $b_1, b_2, \dots, b_k$ . Let $\alpha$ a sequence of size $n$ and your terms are in $[1, n]$ , such that each term appears $h$ times in the sequence $\alpha$ and $2h\geq n$ . Show that $\alpha$ is equivalent to some sequence $\beta$ which contains a subsequence such that your size is(at most) equal to $h$ and your sum is exactly equal to $n$ .