INMO Previous Paper 2017

In the given figure, $$ABCD$$ is a square paper. It is folded along EF such that A goes to a point $$A' \in C,B$$ on the side BC and D goes to $$D'$$. The line $$A'D'$$ cuts $$CD$$ in $$G$$. Show that the inradius of the triangle $$GCA'$$ is the sum of the inradii of the triangles $$GD'F$$ and $$A'BE$$.

Suppose $$n \geq 0$$ is an integer and all the roots of $$x^{3}+\alpha x+4-(2 \times 2016^{n})=0$$ 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 $$\left\{1, 2, 3, 4, 5, 6, 7, 8, 9\right\}$$ such that $$x^{2}-a \left\{x\right\}+b=0$$, where $$\left\{x\right\}$$ denotes the fractional part of the real number $$x$$. (For example $$\left\{1.1\right\}$$ = 0.1 = $$\left\{−0.9\right\}$$.)

Let $$ABCDE$$ 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 $$AB + BC + CD$$.

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 perpendicular to $$PQ$$ and $$AI = PQ$$.

Let $$n \geq 1$$ be an integer and consider the sum $$x=\sum_{k \geq 0}^{}\left(_{2k}^{n}\right)2^{n-2k}3^{k}=\left(_{0}^{n}\right)2^{n}+\left(_{2}^{n}\right)2^{n-2}.3+\left(_{4}^{n}\right)2^{n-4}.3^{2}+....$$ Show that $$2x − 1, 2x, 2x + 1$$ form the sides of a triangle whose area and inradius are also integers.