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.

  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

RMO 2017 Problem 5

Problem 6