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.
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.

Discussion by Writabrata Bhattacharya (Associate Faculty – Cheenta)

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)) = 0Discussion: https://www.cheenta.com/forums/topic/rmo-2017-p3-2/
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 $$1 \le i < j \le n$$ and $$1 \le k < l \le n$$ the thre squares with centers at (i, k), (j, k) , (j, l) have distinct colours. What is the least possible colours needed?
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 $$\Omega$$Discussion by Sauvik Mondal (Faculty – Cheenta)
6. Let x, y, z be real numbers, each greater than 1. Prove that $$\frac {x+1}{y +1 } + \frac {y+1}{z+1} + \frac {z+1}{x+1} \le\frac {x – 1}{y – 1 } + \frac {y- 1}{z-1} + \frac {z-1}{x-1}$$Discussion by Writabrata Bhattacharya (Associate Faculty – Cheenta)

Problem 2 Problem 5  Problem 6 