a) In a $ XYZ$ triangle, the incircle tangents the $ XY $ and $ XZ $ sides at the $ T $ and $ W $ points, respectively. Prove that: $$ XT = XW = \frac {XY + XZ-YZ} {2} $$ Let $ ABC $ be a triangle and $ D $ is the foot of the relative height next to $ A. $ Are $ I $ and $ J $ the incentives from triangle $ ABD $ and $ ACD $, respectively. The circles of $ ABD $ and $ ACD $ tangency $ AD $ at points $ M $ and $ N $, respectively. Let $ P $ be the tangency point of the $ BC $ circle with the $ AB$ side. The center circle $ A $ and radius $ AP $ intersect the height $ D $ at $ K. $ b) Show that the triangles $ IMK $ and $ KNJ $ are congruent c) Show that the $ IDJK $ quad is inscritibed

Let $a, b, c>0$ such that $a^{2}+b^{2}+c^{2}=3abc.$ Prove the following inequality: \[ \frac{a}{b^{2}c^{2}}+\frac{b}{c^{2}a^{2}}+\frac{c}{a^{2}b^{2}}\geq\frac{9}{a+b+c} \]

Find the prime number $p$ for which $p + 2500$ is a perfect square.

Antonio and Bernardo play the following game. They are given two piles of chips, one with $m$ and the other with $n$ chips. Antonio starts, and thereafter they make in turn one of the following moves: (i) take a chip from one pile; (ii) take a chip from each of the piles; (ii) remove a chip from one of the piles and put it onto the other. Who cannot make any more moves, loses. Decide, as a function of $m$ and $n$ if one of the players has a winning strategy, and in the case of the affirmative answer describe that strategy.

Let $k>1$. Fix a circle $\omega$ with center $O$ and radius $r$, and fix a point $A$ with $OA=kr$. Let $AB$, $AC$ be tangents to $\omega$. Choose a variable point $P$ on the minor arc $BC$ in $\omega$. Lines $AB$ and $CP$ intersect at $X$ and lines $AC$ and $BP$ intersect at $Y$. The circles $(BPX)$ and $(CPY)$ meet at another point $Z$. Prove that the line $PZ$ always passes through a fixed point except for one value of $k>1$, and determine this value. [i]Proposed by Ivan Chan Kai Chin[/i]

Let $S$ be the subset of rational numbers that can be written in the form $a/b$, where $a$ is any integer and $b$ is an odd integer. Does the sum of two of its elements belong to the $S$ ? And the product? Are there elements in $S$ whose inverse belongs to $S$ ?

How many integers from $1$ to $2016$ are divisible by $3$ or $7$, but not $21$?

The temperatures $ f^\circ \text{F}$ and $ c^\circ \text{C}$ are equal when $ f \equal{} \frac {9}{5}c \plus{} 32$. What temperature is the same in both $ ^\circ \text{F}$ and $ ^\circ \text{C}$?

A person observes a building of height $h$ at an angle of inclination $\alpha$ from a point on the ground. After walking a distance $a$ towards it, the angle is now $2\alpha$, and walking a further distance $b$ causes it to increase to $3\alpha$. Find $h$ in terms of $a$ and $b$.

Let $ m$ be a positive odd integer, $ m > 2.$ Find the smallest positive integer $ n$ such that $ 2^{1989}$ divides $ m^n \minus{} 1.$