10th APMO 1998

A1.  S is the set of all possible n-tuples (X1, X2, ... , Xn) where each Xi is a subset of {1, 2, ... , 1998}. For each member k of S let f(k) be the number of elements in the union of its n elements. Find the sum of f(k) over all k in S.
A2.  Show that (36m + n)(m + 36n) is not a power of 2 for any positive integers m, n.
A3.  Prove that (1 + x/y)(1 + y/z)(1 + z/x) ≥ 2 + 2(x + y + z)/w for all positive reals x, y, z, where w is the cube root of xyz.
A4.  ABC is a triangle. AD is an altitude. X lies on the circle ABD and Y lies on the circle ACD. X, D and Y are collinear. M is the midpoint of XY and M' is the midpoint of BC. Prove that MM' is perpendicular to AM.
A5.  What is the largest integer divisible by all positive integers less than its cube root.

To avoid possible copyright problems, I have changed the wording, but not the substance, of the problems.

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© John Scholes
12 April 2002