INMO 2013

Let $\Gamma_{1}$ and $\Gamma_{2}$ be two circles touching each other externally at $R$. Let $l_{1}$ be a line which is tangent to $\Gamma_{2}$ at $P$ and passing through the center $O_{1}$ of $\Gamma_{1}$. Similarly, let $l_{2}$ be a line which is tangent to $\Gamma_{2}$ at $Q$ and passing through the center $O_{2}$ of $\Gamma_{2}$. Suppose $l_{1}$ and $l_{2}$ are not parallel and interesct at $K$. If $K P=K Q$, prove that the triangle $P Q R$ is equilateral.

Find all positive integers $m, n$, and primes $p \geq 5$ such that

$$m\left(4 m^{2}+m+12\right)=3\left(p^{n}-1\right)$$

Let $a, b, c, d$ be positive integers such that $a \geq b \geq c \geq d$. Prove that the equation $x^{4}-a x^{3}-b x^{2}-c x-d=0$ has no integer solution.

Let $n$ be a positive integer. Call a nonempty subset $S$ of $\{1,2, \ldots, n\}$ good if the arithmetic mean of the elements of $S$ is also an integer. Further let $t_{n}$ denote the number of good subsets of $\{1,2, \ldots, n\}$. Prove that $t_{n}$ and $n$ are both odd or both even.

In an acute triangle $A B C, O$ is the circumcenter, $H$ is the orthocenter and $G$ is the centroid. Let $O D$ be perpendicular to $B C$ and $H E$ be perpendicular to $C A$, with $D$ on $B C$ and $E$ on $C A$. Let $F$ be the midpoint of $A B$. Suppose the areas of triangles $O D C, H E A$ and $G F B$ are equal. Find all the possible values of $\widehat{C}$.

Let $a, b, c, x, y, z$ be positive real numbers such that $a+b+c=x+y+z$ and abc $=x y z$. Further, suppose that $a \leq x<y<z \leq c$ and $a<b<c$. Prove that $a=x, b=y$ and $c=z$.