## 14 Oct Regional Mathematical Olympiad India 2017 Questions

The Regional Mathematical Olympiad (RMO) is organized every year in India throughout the country as a preliminary screening for the Indian National Mathematical Olympiad (INMO). This year it was held on 8th October, 2017. The questions of this year’s RMO can be found below.

(All questions carry equal marks, maximum possible score is 102.)

1. Let $AOB$ be a given angle less than $180^{\circ}$ and let $P$ be an interior point of the angular region determined by the 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 ray $AO$ and $D$ lies on the $OB$, and $CP:PD=1:2$.
2. Show that the equation $a^3+(a+1)^3+(a+2)^3+(a+3)^3+(a+4)^3+(a+5)^3+(a+6)^3=b^4+(b+1)^4$ has no solutions in integers for $a,b$.
3. Let $P(x)=x^2+\frac{1}{2}x+b$ and $Q(x)=x^3+cx+d$ be two polynomials with real coefficients such that $P(x)Q(x)=Q(P(x))$ for all real $x$. Find all the 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 and $1\leq k, 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 chord $AB$ which is not a diameter. Let $\Gamma_1$ 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 $\frac{x+1}{y+1}+\frac{y+1}{z+1}+\frac{z+1}{x+1} \leq \frac{x-1}{y-1}+\frac{y-1}{z-1}+\frac{z-1}{x-1}$.

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