Best Man Mathematical Olympiad

This Olympiad consists of 5 problems, with the fifth problem for tie breaking only. The time limit is 1 day. The winner isthe contestant with the maximum sum of scores on problems 1 through 4. Ties will be broken by scores on problem 5. If tiesstill exist, they will further be broken by overall solution quality.

Saturday, May 16, 2015

Problem 1. Find all z C satisfying

(z z2)(1 z + z2)2 = 17

Problem 2. Find the maximum size of a subset S of {1, 2, . . . , 31} with the property that no two distinct elements of S sumto a perfect square.

Problem 3. Over all points P coplanar to square ABCD, find the maximum and minimum values of

|PA|+ |PB||PC|+ |PD|

Problem 4. Let a, b, c be positive integers such that a b c and ac2 is prime. Prove that if

a2 + b2 + c2 2(ab + bc + ca) = b,then b is either prime or a perfect square.

Problem 5. Let A be a finite ring such that 1 + 1 = 0. Prove that the equations x2 = 0 and x2 = 1 have the same numberof solutions in A.

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