 1. In the given figure, $ABCD$ is a square sheet of paper. It is folded along $E F$ such that $A$ goes to a point $A’$ different from Band $C$, on the side $BC$ and $D$ goes to $D’$ . The line $A’ D’$ cuts $C D$ in $G$. Show that the inradius of the triangle $GC A’$ is the sum of the inradii of the triangles $GD’F$ and $A’ BE$. 2. Suppose $n \ge 0$ is an integer and all the roots of $x^3 + ax + 4 -(2 \times {2016^n})$ = 0 are integers. Find all possible values of $\alpha$.
3. 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 =$\{-0.9\}$ ).
4. Let $ABCDE$ be a convex pentagon in which ${\angle A} ={\angle B} ={\angle C} ={\angle D}$ =$120^{\circ}$ and side lengths are five consecutive integers in some order. Find all possible values of  $AB + BC + CD$.
5. Let $ABC$ be a triangle with $\angle A =90^{\circ}$ and $AB < AC$. Let $AD$ be the altitude from $A$ on to BC. Let $P, Q$ and I denote respectively the incentres of triangles $ABD, ACD$ and $ABC$. Prove that $AI$ is prependicular to $PQ$ and $AI = PQ$.
6. Let $n \ge 1$ be an integer and consider the sum
$\displaystyle{ x ={\sum_{k \ge 0}} \dbinom{n}{2k} 2{^{n-2k}} {3^k} \\ = \dbinom{n}{0} 2^n + \dbinom{n}{2} 2{^{n-2}} . 3 + \dbinom{n}{4} 2{^{n-4}} . 3^2 + \dots }$
Show that $2x – 1, 2x, 2x+1$ from the sides of a triangle whose area and inradius are also integers.