# [Solutions] India National Mathematical Olympiad 2016

1. Let $ABC$ be a triangle in which $AB=AC$. Suppose the orthocentre of the triangle lies on the incircle. Find the ratio $\frac{AB}{BC}$.
2. For positive real numbers $a,b,c$ which of the following statements necessarily implies $a=b=c$
a) $a(b^3+c^3)=b(c^3+a^3)=c(a^3+b^3)$,
b) $a(a^3+b^3)=b(b^3+c^3)=c(c^3+a^3)$.
3. Let $\mathbb{N}$ denote the set of natural numbers. Define a function $T:\mathbb{N}\rightarrow\mathbb{N}$ by $T(2k)=k$ and $T(2k+1)=2k+2$. We write $T^2(n)=T(T(n))$ and in general $T^k(n)=T^{k-1}(T(n))$ for any $k>1$.
a) Show that for each $n\in\mathbb{N}$, there exists $k$ such that $T^k(n)=1$.
b) For $k\in\mathbb{N}$, let $c_k$ denote the number of elements in the set $\{n: T^k(n)=1\}$. Prove that $c_{k+2}=c_{k+1}+c_k$, for $k\ge 1$.
4. Suppose $2016$ points of the circumference of a circle are colored red and the remaining points are colored blue . Given any natural number $n\ge 3$, prove that there is a regular $n$-sided polygon all of whose vertices are blue.
5. Let $ABC$ be a right-angle triangle with $\angle B=90^{\circ}$. Let $D$ be a point on $AC$ such that the inradii of the triangles $ABD$ and $CBD$ are equal. If this common value is $r^{\prime}$ and if $r$ is the inradius of triangle $ABC$. Prove that $\cfrac{1}{r'}=\cfrac{1}{r}+\cfrac{1}{BD}.$
6. Consider a nonconstant arithmetic progression $a_1, a_2,\cdots, a_n,\cdots$. Suppose there exist relatively prime positive integers $p>1$ and $q>1$ such that $a_1^2, a_{p+1}^2$ and $a_{q+1}^2$ are also the terms of the same arithmetic progression. Prove that the terms of the arithmetic progression are all integers.
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