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APMO 1998

Problems

Problem 1: Let F be the set of all n-tuples (A1, A2,,An) where each Ai, i = 1,2,,n is a subset of {1,2,,1998}. Let |A| denote the number of elements of the set A. Find the number

.

 

Problem 2: Show that for any positive integers a and b, (36a + b)(a + 36b) cannot be a power of 2.

 

Problem 3: Let a, b, c be positive real numbers. Prove that

 

Problem 4: Let ABC be a triangle and D the foot of the altitude from A. Let E and F be on a line passing though D such that AE is perpendicular to BE, AF is perpendicular to CF, and E and F are different from D. Let M and N be the midpoints of the line segments BC and EF, respectively. Prove that AN is perpendicular to NM.

 

Problem 5: Determine the largest of all integers n with the property that n is divisible by all positive integers that are less than .