Regional Math Olympiad 2017

Here are the questions asked in Regional Math Olympiad 2017 and their solutions. Try to solve it first and then see the solutions.

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RMO 2017, Problem 1:

Let AOB be a given angle less than $ 180^o $ 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 ray OA and D lies on the ray OB and CP:PD = 1:2.

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RMO 2017, Problem 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 integer a, b.

Written solution

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RMO 2017, Problem 3:

Let $P(x) = x^2 + \frac {1}{2} x + b$ 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

RMO 2017, Problem 4:

Consider $n^2$ unit squares in the xy-plane centered at the point (i, j) with integer coordinates, $1 \le i \le n$ , $1 \le j \le n$ . It is required to color each unit square in such a way that whenever 1≤ i < j and 1 ≤ k < l ≤ n the three squares with centers at (i, k), (j, k) , (j, l) have distinct colours. What is the least possible colours needed?

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RMO 2017, Problem 5:

Let $\Omega$ be a circle with a 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 Ω

Written solution part 1

Written solution part 2

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RMO 2017, Problem 6:

Let x, y, z be real numbers, each greater than 1. Prove that

$\frac{x+1}{y+1}$ + $\frac{y+1}{z+1}$ + $\frac{z+1}{x+1}$ $\leq$ $\frac{x-1}{y-1}$ + $\frac{y-1}{z-1}$ + $\frac{z-1}{x-1}$.

Written solution

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~ Discussion by Souvik Mondal & Writabrata Bhattacharya (Associate Faculty - Cheenta)

Some Useful Links:

Practice Previous Year Problems for RMO
Our Math Olympiad Program
Cyclic Pentagon - RMO 2008, Problem 1 - Watch and Learn

Let AOB be a given angle less than $ 180^o $ 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 ray OA and D lies on the ray OB and CP:PD = 1:2.

Watch

RMO 2017, Problem 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 integer a, b.

Written solution

Watch

RMO 2017, Problem 3:

Let $P(x) = x^2 + \frac {1}{2} x + b$ 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

RMO 2017, Problem 4:

Consider $n^2$ unit squares in the xy-plane centered at the point (i, j) with integer coordinates, $1 \le i \le n$ , $1 \le j \le n$ . It is required to color each unit square in such a way that whenever 1≤ i < j and 1 ≤ k < l ≤ n the three squares with centers at (i, k), (j, k) , (j, l) have distinct colours. What is the least possible colours needed?

Watch

RMO 2017, Problem 5:

Let $\Omega$ be a circle with a 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 Ω