Indian National Mathematics Olympiad 2002 Problems

Indian National Mathematics Olympiad 2002 Problems

1.  ABCDEF is a convex hexagon. Consider the following statements. (1) AB is parallel to DE, (2) BC is parallel to EF, (3) CD is parallel to FA, (4) AE = BD, (5) BF = CE, (6) CA = DF. Show that if any five of these statements are true then the hexagon is cyclic.

2.  Find the smallest positive value taken by a³ + b³ + c³ - 3abc for positive integers a, b, c. Find all a, b, c which give the smallest value.

3.  x, y are positive reals such that x + y = 2. Show that x³y³(x³ + y³) ≤ 2.

4.  Do there exist 100 lines in the plane, no three concurrent, such that they intersect in exactly 2002 points?

5.  Do there exist distinct positive integers a, b, c such that a, b, c, -a+b+c, a-b+c, a+b-c, a+b+c form an arithmetic progression (in some order).

6.  The numbers 1, 2, 3, ... , n² are arranged in an n x n array, so that the numbers in each row increase from left to right, and the numbers in each column increase from top to bottom. Let a_ij be the number in position i, j. Let b_j be the number of possible value for a_jj. Show that b₁ + b₂ + ... + b_n = n(n²-3n+5)/3.