INMO 2017

In the given figure, $A B C D$ is a square paper. It is folded along $E F$ such that $A$ goes to a point $A^{\prime} \neq C, B$ on the side $B C$ and $D$ goes to $D^{\prime}$. The line $A^{\prime} D^{\prime}$ cuts $C D$ in $G$. Show that the inradius of the triangle $G C A^{\prime}$ is the sum of the inradii of the triangles $G D^{\prime} F$ and $A^{\prime} B E$.

Suppose $n \geq 0$ is an integer and all the roots of $x^{3}+\alpha x+4-\left(2 \times 2016^{n}\right)=0$ are integers. Find all possible values of $\alpha$.

Find the number of triples $(x, a, b)$ where $x$ is a real number and $a, b$ belong to the set $\{1,2,3,4,5,6,7,8,9\}$ such that

$$x^{2}-a\{x\}+b=0$$

where $\{x\}$ denotes the fractional part of the real number $x$. (For example $\{1.1\}=0.1=$ $\{-0.9\}$.

Let $A B C D E$ be a convex pentagon in which $\angle A=\angle B=\angle C=\angle D=120^{\circ}$ and whose side lengths are 5 consecutive integers in some order. Find all possible values of $A B+B C+C D$.

Let $A B C$ be a triangle with $\angle A=90^{\circ}$ and $A B<A C$. Let $A D$ be the altitude from $A$ on to $B C$. Let $P, Q$ and $I$ denote respectively the incentres of triangles $A B D, A C D$ and $A B C$. Prove that $A I$ is perpendicular to $P Q$ and $A I=P Q$.

Let $n \geq 1$ be an integer and consider the sum

$$x=\sum_{k \geq 0}\binom{n}{2 k} 2^{n-2 k} 3^{k}=\binom{n}{0} 2^{n}+\binom{n}{2} 2^{n-2} \cdot 3+\binom{n}{4} 2^{n-4} \cdot 3^{2}+\cdots$$

Show that $2 x-1,2 x, 2 x+1$ form the sides of a triangle whose area and inradius are also integers.