Difference between revisions of "1999 USAMO Problems"
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Revision as of 18:12, 17 September 2012
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 |