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 and Q(x)=x2+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 unit squares in the xy-plane centered at the point (i, j) with integer coordinates, 1≤i≤n , 1≤j≤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 be a circle with a chord AB which is not a diameter. Let Γ1 be a circle on one side of AB such that it is tangent to AB at C and internally tangent to Ω at D. Likewise let Γ2 be a circle on the other side of AB such that it is tangent to AB at E and internally tangent to Ω at F. Suppose the line DC intersects Ω at X≠D and the line FE intersects Ω at Y≠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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