India National Olympiad 2017 Problems
Problem 1
In the given figure, $ABCD$ is a square sheet of paper. It is folded along $EF$ such that $A$ goes to a point $A’$ different from $B$ and $C$, 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 $A’BE$.
Problem 2
Suppose $n \ge 0$ 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 $\alpha$.
Problem 3
Find the number of triples $(x, a, b)$ where $x$ is a real numbe 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 $\{1.1\} = .1 = \{-0.9\}$.)
Problem 4
Let $ABCDE$ be a convex pentagon in which $\angle{A}=\angle{B}=\angle{C}=\angle{D}=120^{\circ}$ and the side lengths are five consecutive integers in some order. Find all possible values of $AB+BC+CD$.
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 triangle $ABD,ACD$ and $ABC$. Prove that $AI$ is perpendicular to $PQ$ and $AI=PQ$.
Problem 6
Let $n\ge 1$ be an integer and consider the sum$$x=\sum_{k\ge 0} \dbinom{n}{2k} 2^{n-2k}3^k=\dbinom{n}{0}2^n+\dbinom{n}{2}2^{n-2}\cdot{}3+\dbinom{n}{4}2^{n-k}\cdot{}3^2 + \cdots{}.$$Show that $2x-1,2x,2x+1$ form the sides of a triangle whose area and inradius are also integers.
