# [Solutions] India Regional Mathematical Olympiad 2017

1. Let $AOB$ be a given angle less than $180^{\circ}$ and let $P$ be an interior point of the angular region determined by $\angle AOB$. Show, with proof, how to construct, using only ruler and compass, a line segment $CD$ passing through $P$ such that $C$ lies on the way $OA$ and $D$ lies on the ray $OB$, and $CP:PD=1:2$.
2. Show that the equation $a^3+(a+1)^3+\ldots+(a+6)^3=b^4+(b+1)^4$ has no solutions in integers $a,b$.
3. Let $P(x)=x^2+\dfrac x 2 +b$ and $Q(x)=x^2+cx+d$ be two polynomials with real coefficients such that $P(x)Q(x)=Q(P(x))$ for all real $x$. Find all real roots of $P(Q(x))=0$.
4. Consider $n^2$ unit squares in the $xy$ plane centered at point $(i,j)$ with integer coordinates, $1 \leq i \leq n$, $1 \leq j \leq n$. It is required to colour each unit square in such a way that whenever $1 \leq i < j \leq n$ and $1 \leq k < l \leq n$, the three squares with centres at $(i,k),(j,k),(j,l)$ have distinct colours. What is the least possible number of colours needed?
5. Let $\Omega$ be a circle with a chord $AB$ which is not a diameter. $\Gamma_{1}$ be a circle on one side of $AB$ such that it is tangent to $AB$ at $C$ and internally tangent to $\Omega$ at $D$. Likewise, let $\Gamma_{2}$ be a circle on the other side of $AB$ such that it is tangent to $AB$ at $E$ and internally tangent to $\Omega$ at $F$. Suppose the line $DC$ intersects $\Omega$ at $X \neq D$ and the line $FE$ intersects $\Omega$ at $Y \neq F$. Prove that $XY$ is a diameter of $\Omega$ .
6. Let $x,y,z$ be real numbers, each greater than $1$. Prove that $$\dfrac{x+1}{y+1}+\dfrac{y+1}{z+1}+\dfrac{z+1}{x+1} \leq \dfrac{x-1}{y-1}+\dfrac{y-1}{z-1}+\dfrac{z-1}{x-1}.$$
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