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..