1999 USAMO Problems
Contents
Day 1
Problem 1
Some checkers placed on an checkerboard satisfy the following conditions:
(a) every square that does not contain a checker shares a side with one that does;
(b) given any pair of squares that contain checkers, there is a sequence of squares containing checkers, starting and ending with the given squares, such that every two consecutive squares of the sequence share a side.
Prove that at least checkers have been placed on the board.
Problem 2
Let be a cyclic quadrilateral. Prove that
Problem 3
Let be a prime and let be integers not divisible by , such that for any integer not divisible by . Prove that at least two of the numbers , , , , , are divisible by . (Note: denotes the fractional part of .)
Day 2
Problem 4
Let () be real numbers such that Prove that .
Problem 5
The Y2K Game is played on a grid as follows. Two players in turn write either an S or an O in an empty square. The first player who produces three consecutive boxes that spell SOS wins. If all boxes are filled without producing SOS then the game is a draw. Prove that the second player has a winning strategy.
Problem 6
Let be an isosceles trapezoid with . The inscribed circle of triangle meets at . Let be a point on the (internal) angle bisector of such that . Let the circumscribed circle of triangle meet line at and . Prove that the triangle is isosceles.
See Also
1999 USAMO (Problems • Resources) | ||
Preceded by 1998 USAMO |
Followed by 2000 USAMO | |
1 • 2 • 3 • 4 • 5 • 6 | ||
All USAMO Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.