# [Solutions] India National Mathematical Olympiad 2010

1. Let $ABC$ be a triangle with circum-circle $\Gamma$. Let $M$ be a point in the interior of triangle $ABC$ which is also on the bisector of $\angle A$. Let $AM, BM, CM$ meet $\Gamma$ in $A_{1}, B_{1}, C_{1}$ respectively. Suppose $P$ is the point of intersection of $A_{1}C_{1}$ with $AB$; and $Q$ is the point of intersection of $A_{1}B_{1}$ with $AC$. Prove that $PQ$ is parallel to $BC$.
2. Find all natural numbers $n > 1$ such that $n^{2}$ does $\text{not}$ divide $(n - 2)!$.
3. Find all non-zero real numbers $x, y, z$ which satisfy the system of equations $(x^2 + xy + y^2)(y^2 + yz + z^2)(z^2 + zx + x^2) = xyz$ $(x^4 + x^2y^2 + y^4)(y^4 + y^2z^2 + z^4)(z^4 + z^2x^2 + x^4) = x^3y^3z^3$
4. How many 6-tuples $(a_1,a_2,a_3,a_4,a_5,a_6)$ are there such that each of $a_1,a_2,a_3,a_4,a_5,a_6$ is from the set $\{1,2,3,4\}$ and the six expressions $a_j^2 - a_ja_{j + 1} + a_{j + 1}^2$ for $j = 1,2,3,4,5,6$ (where $a_7$ is to be taken as $a_1$) are all equal to one another?
5. Let $ABC$ be an acute-angled triangle with altitude $AK$. Let $H$ be its ortho-centre and $O$ be its circum-centre. Suppose $KOH$ is an acute-angled triangle and $P$ its circum-centre. Let $Q$ be the reflection of $P$ in the line $HO$. Show that $Q$ lies on the line joining the mid-points of $AB$ and $AC$.
6. Define a sequence $\{a_n \}_{n\geq0}$ by $$a_0 = 0,\, a_1 = 1,\quad a_n = 2a_{n - 1} + a_{n - 2},\,\forall n\geq2.$$ a) For every $m > 0$ and $0\leq j\leq m,$ prove that $2a_m$ divides $a_{m + j} + ( - 1)^ja_{m - j}$.
b) Suppose $2^k$ divides $n$ for some natural numbers $n$ and $k$. Prove that $2^k$ divides $a_n.$
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