Regional Mathematical Olympiad India 2017 Questions

The Regional Mathematical Olympiad (RMO) is organized every year in India throughout the country as a preliminary screening for the Indian National Mathematical Olympiad (INMO). This year it was held on 8th October, 2017. The questions of this year’s RMO can be found below.

(All questions carry equal marks, maximum possible score is 102.)

  1. Let AOB be a given angle less than 180^{\circ} and let P be an interior point of the angular region determined by the 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 AO and D lies on the 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 integers for a,b.
  3. Let P(x)=x^2+\frac{1}{2}x+b and Q(x)=x^3+cx+d be two polynomials with real coefficients such that P(x)Q(x)=Q(P(x)) for all real x. Find all the real roots of P(Q(x))=0.
  4. Consider n^2 unit squares in the xy-plane centered at point (i,j) with integer coordinates, 1\leq i\leq n, 1\leq j \leq n. It is required to colour each unit square in such a way that whenever 1\leq i<j\leq n and 1\leq k<l\leq n, the three squares with centres at (i,k),(j,k),(j,l) have distinct colours. What is the least possible number of colours needed?
  5. Let \Omega be a circle with 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.
  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}.