1989 USAMO Problems
Problems from the 1989 USAMO.
For each positive integer , let
Find, with proof, integers such that and .
The 20 members of a local tennis club have scheduled exactly 14 two-person games among themselves, with each member playing in at least one game. Prove that within this schedule there must be a set of 6 games with 12 distinct players.
Let be a polynomial in the complex variable , with real coefficients . Suppose that . Prove that there exist real numbers and such that and .
Let be an acute-angled triangle whose side lengths satisfy the inequalities . If point is the center of the inscribed circle of triangle and point is the center of the circumscribed circle, prove that line intersects segments and .
Let and be real numbers such that
Determine, with proof, which of the two numbers, or , is larger.
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