# [Solutions] India National Mathematical Olympiad 2008

1. Let $ABC$ be triangle, $I$ its in-center; $A_1,B_1,C_1$ be the reflections of $I$ in $BC, CA, AB$ respectively. Suppose the circum-circle of triangle $A_1B_1C_1$ passes through $A$. Prove that $B_1,C_1,I,I_1$ are concylic, where $I_1$ is the in-center of triangle $A_1,B_1,C_1$.
2. Find all triples $\left(p,x,y\right)$ such that $p^x=y^4+4$, where $p$ is a prime and $x$ and $y$ are natural numbers.
3. Let $A$ be a set of real numbers such that $A$ has at least four elements. Suppose $A$ has the property that $a^2 + bc$ is a rational number for all distinct numbers $a,b,c$ in $A$. Prove that there exists a positive integer $M$ such that $a\sqrt{M}$ is a rational number for every $a$ in $A$.
4. All the points with integer coordinates in the $xy$-Plane are coloured using three colours, red, blue and green, each colour being used at least once. It is known that the point $(0,0)$ is red and the point $(0,1)$ is blue. Prove that there exist three points with integer coordinates of distinct colours which form the vertices of a right-angled triangle.
5. Let $ABC$ be a triangle; $\Gamma_A,\Gamma_B,\Gamma_C$ be three equal, disjoint circles inside $ABC$ such that $\Gamma_A$ touches $AB$ and $AC$; $\Gamma_B$ touches $AB$ and $BC$; and $\Gamma_C$ touches $BC$ and $CA$. Let $\Gamma$ be a circle touching circles $\Gamma_A, \Gamma_B, \Gamma_C$ externally. Prove that the line joining the circum-centre $O$ and the in-centre $I$ of triangle $ABC$ passes through the centre of $\Gamma$.
6. Let $P(x)$ be a polynomial with integer coefficients. Prove that there exist two polynomials $Q(x)$ and $R(x)$, again with integer coefficients, such that
• $P(x) \cdot Q(x)$ is a polynomial in $x^2$ , and
• $P(x) \cdot R(x)$ is a polynomial in $x^3$.
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