1984 USAMO Problems
Problem 1
The product of two of the four roots of the quartic equation is
. Determine the value of
.
Problem 2
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?
Problem 3
and
are five distinct points in space such that
, where
is a given acute angle. Determine the greatest and least values of
.
Problem 4
A dfficult 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.
Problem 5
is a polynomial of degree
such that
Determine .
See Also
1984 USAMO (Problems • Resources) | ||
Preceded by 1983 USAMO |
Followed by 1985 USAMO | |
1 • 2 • 3 • 4 • 5 | ||
All USAMO Problems and Solutions |