Indian National Mathematics Olympiad 2004 Problems

Indian National Mathematics Olympiad 2004 Problems

1.  ABCD is a convex quadrilateral. K, L, M, N are the midpoints of the sides AB, BC, CD, DA. BD bisects KM at Q. QA = QB = QC = QD, and LK/LM = CD/CB. Prove that ABCD is a square.

2.  p > 3 is a prime. Find all integers a, b, such that a² + 3ab + 2p(a+b) + p² = 0.

3.  If α is a real root of x⁵ - x³ + x - 2 = 0, show that [α⁶] = 3.

4.  ABC is a triangle, with sides a, b, c (as usual), circumradius R, and exradii r_a, r_b, r_c. If 2R ≤ r_a, show that a > b, a > c, 2R > r_b, and 2R > r_c.

5.  S is the set of all (a, b, c, d, e, f) where a, b, c, d, e, f are integers such that a² + b² + c² + d² + e² = f². Find the largest k which divides abcdef for all members of S.

6.  Show that the number of 5-tuples (a, b, c, d, e) such that abcde = 5(bcde + acde + abde + abce + abcd) is odd.