OMM 2016

Let $C_1$ and $C_2$ be two circumferences externally tangents at $S$ such that the radius of $C_2$ is the triple of the radius of $C_1$ . Let a line be tangent to $C_1$ at $P \neq S$ and to $C_2$ at $Q \neq S$ . Let $T$ be a point on $C_2$ such that $QT$ is diameter of $C_2$ . Let the angle bisector of $\angle SQT$ meet $ST$ at $R$ . Prove that $QR=RT$

A pair of positive integers $m, n$ is called guerrera, if there exists positive integers $a, b, c, d$ such that $m=ab$ , $n=cd$ and $a+b=c+d$ . For example the pair $8, 9$ is guerrera cause $8= 4 \cdot 2$ , $9= 3 \cdot 3$ and $4+2=3+3$ . We paint the positive integers if the following order:

We start painting the numbers $3$ and $5$ . If a positive integer $x$ is not painted and a positive $y$ is painted such that the pair $x, y$ is guerrera, we paint $x$ .

Find all positive integers $x$ that can be painted.

Find the minimum real $x$ that satisfies $$\lfloor x \rfloor <\lfloor x^2 \rfloor <\lfloor x^3 \rfloor < \cdots < \lfloor x^n \rfloor < \lfloor x^{n+1} \rfloor < \cdots$$

The numbers from $1$ to $n^2$ are written in order in a grid of $n \times n$ , one number in each square, in such a way that the first row contains the numbers from $1$ to $n$ from left to right; the second row contains the numbers $n + 1$ to $2n$ from left to right, and so on and so forth. An allowed move on the grid consists in choosing any two adjacent squares (i.e. two squares that share a side), and add (or subtract) the same integer to both of the numbers that appear on those squares.

Find all values of $n$ for which it is possible to make every squares to display $0$ after making any number of moves as necessary and, for those cases in which it is possible, find the minimum number of moves that are necessary to do this.

Let $ABCD$ a quadrilateral inscribed in a circumference, $l_1$ the parallel to $BC$ through $A$ , and $l_2$ the parallel to $AD$ through $B$ . The line $DC$ intersects $l_1$ and $l_2$ at $E$ and $F$ , respectively. The perpendicular to $l_1$ through $A$ intersects $BC$ at $P$ , and the perpendicular to $l_2$ through $B$ cuts $AD$ at $Q$ . Let $\Gamma_1$ and $\Gamma_2$ be the circumferences that pass through the vertex of triangles $ADE$ and $BFC$ , respectively. Prove that $\Gamma_1$ and $\Gamma_2$ are tangent to each other if and only if $DP$ is perpendicular to $CQ$ .