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.