CN-LEAGUE 2017

Given an isocleos triangle $ABC$ with equal sides $AB=AC$ and incenter $I$ .Let $\Gamma_1$ be the circle centered at $A$ with radius $AB$ , $\Gamma_2$ be the circle centered at $I$ with radius $BI$ .A circle $\Gamma_3$ passing through $B,I$ intersects $\Gamma_1$ , $\Gamma_2$ again at $P,Q$ (different from $B$ ) respectively.Let $R$ be the intersection of $PI$ and $BQ$ .Show that $BR \perp CR$ .

Each square of a $33\times 33$ square grid is colored in one of the three colors: red, yellow or blue, such that the numbers of squares in each color are the same. If two squares sharing a common edge are in different colors, call that common edge a separating edge. Find the minimal number of separating edges in the grid.

Let $m,n$ be integers greater than 1, $m \geq n$ , $a_1,a_2,\dots,a_n$ are $n$ distinct numbers not exceed $m$ ,which are relatively primitive.Show that for any real $x$ ,there exists $i$ for which $||a_ix|| \geq \frac{2}{m(m+1)} ||x||$ ,where $||x||$ denotes the distance between $x$ and the nearest integer to $x$ .

Let $x_1,x_2,x_3\geq 0$ and $x_1+x_2+x_3=1$ . Find the minimum value and the maximum value of $(x_1+3x_2+5x_3)\left(x_1+\frac{x_2}{3}+\frac{x_3}{5}\right).$