1989 USAMO Problems
Problem 1
For each positive integer , let
.
Find, with proof, integers such that
and
.
Problem 2
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.
Problem 3
Let be a polynomial in the complex variable
, with real coefficients
. Suppose that
. Prove that there exist real numbers
and
such that
and
.
Problem 4
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
.