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# INMO 2016

1. Let $ABC$ be triangle in which $AB=AC$. Suppose the orthocentre of the triangle lies on the incircle. Find the ratio $AB/BC$.
2. For positive real numbers $a, b, c,$ which of the following statements necessarily implics $a= b= c:$ (I) $a (b^3+c^3)$ = $b {(c^3 +a3)}$ = $c {(a^3 +b^3)}$,
(II) $a {(a^3 + b^3)}$ = $b {(b^3 + c^3)}$ = $c {(c^3 + a^3)}$ ? Justify your answer.
3. Let $N$ denote the set of all natural numbers. Define a function $T : N \rightarrow {N}$ 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.(i) Show that for each $n $$\in$ N$$, there exists $k$ such thst  ${T^k} {(n)} =1$.(ii) For $k\in$ 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 coloured red and the remaining points are coloured 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-angled triangle with ${\angle B} = 90^{\circ}$ . Let $D$ be a point on $AC$ such that the inradii of the triangles $ABD$ and $C B D$ are equal. If this common value is r’ and if r is the inradius of triangle $ABC$, prove that
$\frac {1} {r’}$= $\frac {1} {r}$ +$\frac {1} {BD}$.
6. Consider a nonconstant arithmetic progression ${a_1}, {a_2},….{ a_n}$,….. Suppose there exist relatively prime positive integers $p > 1\;{\textbf{and}}\; q > 1$ 1 such that ${a^2_1}$, $a^2_{p+1}$ and $a^2_{q+1}$ are also the terms of the same arithmetic progression. Prove that the terms of the arithmetic progression are all integers.