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# 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.

Looking for just the problems? Download the PDF here.

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.

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.

RMO 2017, Problem 3:

Let P(x)=x2+12x+b$P(x) = x^2 + \frac {1}{2} x + b$ and Q(x)=x2+cx+d$Q(x) = x^2 + cx + d$ 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 n2$n^2$ unit squares in the xy-plane centered at the point (i, j) with integer coordinates, 1in$1 \le i \le n$ , 1jn$1 \le j \le n$ . 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?

RMO 2017, Problem 5:

Let Ω$\Omega$ be a circle with a chord AB which is not a diameter. Let Γ1$\Gamma_1$ be a circle on one side of AB such that it is tangent to AB at C and internally tangent to Ω$\Omega$ at D. Likewise let Γ2$\Gamma_2$ be a circle on the other side of AB such that it is tangent to AB at E and internally tangent to Ω$\Omega$ at F. Suppose the line DC intersects Ω$\Omega$ at XD$X \neq D$ and the line FE intersects Ω$\Omega$ at YF$Y \neq F$. Prove that XY is a diameter of Ω

RMO 2017, Problem 6:

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

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

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

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

Looking for just the problems? Download the PDF here.

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.

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.

RMO 2017, Problem 3:

Let P(x)=x2+12x+b$P(x) = x^2 + \frac {1}{2} x + b$ and Q(x)=x2+cx+d$Q(x) = x^2 + cx + d$ 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 n2$n^2$ unit squares in the xy-plane centered at the point (i, j) with integer coordinates, 1in$1 \le i \le n$ , 1jn$1 \le j \le n$ . 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?

RMO 2017, Problem 5:

Let Ω$\Omega$ be a circle with a chord AB which is not a diameter. Let Γ1$\Gamma_1$ be a circle on one side of AB such that it is tangent to AB at C and internally tangent to Ω$\Omega$ at D. Likewise let Γ2$\Gamma_2$ be a circle on the other side of AB such that it is tangent to AB at E and internally tangent to Ω$\Omega$ at F. Suppose the line DC intersects Ω$\Omega$ at XD$X \neq D$ and the line FE intersects Ω$\Omega$ at YF$Y \neq F$. Prove that XY is a diameter of Ω

RMO 2017, Problem 6:

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

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

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

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