1985 USAMO Problems
Problems from the 1985 USAMO.
Determine whether or not there are any positive integral solutions of the simultaneous equations with distinct integers .
Determine each real root of
correct to four decimal places.
Let denote four points in space such that at most one of the distances is greater than . Determine the maximum value of the sum of the six distances.
There are people at a party. Prove that there are two people such that, of the remaining people, there are at least of them, each of whom knows both or else knows neither of the two. Assume that "know" is a symmetrical relation; denotes the greatest integer less than or equal to .
Let be a non-decreasing sequence of positive integers. For , define , that is, is the minimum value of such that . If , determine the maximum value of .
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The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.