1984 USAMO Problems
Problems from the 1984 USAMO.
The product of two of the four roots of the quartic equation is . Determine the value of .
The geometric mean of any set of non-negative numbers is the -th root of their product.
For which positive integers is there a finite set of distinct positive integers such that the geometric mean of any subset of is an integer?
Is there an infinite set of distinct positive integers such that the geometric mean of any finite subset of is an integer?
, , , , and are five distinct points in space such that , where is a given acute angle. Determine the greatest and least values of .
A difficult mathematical competition consisted of a Part I and a Part II with a combined total of problems. Each contestant solved problems altogether. For each pair of problems, there were exactly two contestants who solved both of them. Prove that there was a contestant who, in Part I, solved either no problems or at least four problems.
is a polynomial of degree such that
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