Indian National Mathematics Olympiad 1996 Problems
1. Given any positive integer n, show that there are distinct positive integers a, b such that a + k divides b + k for k = 1, 2, ... , n. If a, b are positive integers such that a + k divides b + k for all positive integers k, show that a = b.
2. C, C' are concentric circles with radii R, 3R respectively. Show that the orthocenter of any triangle inscribed in C must lie inside the circle C'. Conversely, show that any point inside C' is the orthocenter of some circle inscribed in C.
3. Find reals a, b, c, d, e such that 3a = (b + c + d)3, 3b = (c + d + e)3, 3c = (d + e + a)3, 3d = (e + a + b)3, 3e = (a + b + c)3.
4. X is a set with n elements. Find the number of triples (A, B, C) such that A, B, C are subsets of X, A is a subset of B, and B is a proper subset of C.
5. The sequence a1, a2, a3, ... is defined by a1 = 1, a2 = 2, an+2 = 2an+1 - an + 2. Show that for any m, amam+1 is also a term of the sequence.
6. A 2n x 2n array has each entry 0 or 1. There are just 3n 0s. Show that it is possible to remove all the 0s by deleting n rows and n columns.
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