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# Indian National Math Olympiad, INMO 2017 Problems

This post contains problems from Indian National Mathematics Olympiad, INMO 2017. Try them and share your solution in the comments.

INMO 2017, Problem 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 $A' BE$.

INMO 2017, Problem 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 $\alpha$.

INMO 2017, Problem 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 =$\{-0.9\}$ ).

INMO 2017, Problem 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  $AB + BC + CD$.

INMO 2017, Problem 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 $AI = PQ$.

INMO 2017, Problem 6

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