1999 USAMO
USAMO 1999
Day 1 April 27th
1 Some checkers placed on an n × n 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 (n2 − 2)/3 checkers have been placed on the board.
2 Let ABCD be a cyclic quadrilateral. Prove that
|AB − CD| + |AD − BC| ≥ 2|AC − BD|.
3 Let p > 2 be a prime and let a, b, c, d be integers not divisible by p, such that
ra rb rc rd
+ + + =2
p p p p
for any integer r not divisible by p. Prove that at least two of the numbers
a + b, a + c, a + d, b + c, b + d, c + d are divisible by p.
(Note: {x} = x − ⌊x⌋ denotes the fractional part of x.)
Day 2 April 27th
4 Let a1 , a2 , . . . , an (n > 3) be real numbers such that
a1 + a2 + · · · + an ≥ n and a21 + a22 + · · · + a2n ≥ n2 .
Prove that max(a1 , a2 , . . . , an ) ≥ 2.
5 The Y2K Game is played on a 1 × 2000 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.
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Contributors: MithsApprentice, tetrahedr0n, rrusczyk
1999 USAMO
6 Let ABCD be an isosceles trapezoid with AB k CD. The inscribed circle ω of
triangle BCD meets CD at E. Let F be a point on the (internal) angle bisector
of ∠DAC such that EF ⊥ CD. Let the circumscribed circle of triangle ACF
meet line CD at C and G. Prove that the triangle AF G is isosceles.
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These problems are copyright c Mathematical Association of America (http:
//maa.org).
www.artofproblemsolving.com/community/c4497
Contributors: MithsApprentice, tetrahedr0n, rrusczyk
