1999 USAMO

Part of USAMO

Subcontests

(6)

1

Isos Trap

ABCD

be an isosceles trapezoid with

AB \parallel CD

. The inscribed circle

\omega

BCD

CD

E

F

be a point on the (internal) angle bisector of

\angle DAC

EF \perp CD

. Let the circumscribed circle of triangle

ACF

CD

C

G

. Prove that the triangle

AFG

1

Y2K Game

The Y2K Game is played on a

1 \times 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.

1

One of USAMO's hardest NT problems

p > 2

be a prime and let

a,b,c,d

be integers not divisible by

p

\left\{ \dfrac{ra}{p} \right\} + \left\{ \dfrac{rb}{p} \right\} + \left\{ \dfrac{rc}{p} \right\} + \left\{ \dfrac{rd}{p} \right\} = 2

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

\{x\} = x - \lfloor x \rfloor

denotes the fractional part of

x

1

Cyclic Quad

ABCD

be a cyclic quadrilateral. Prove that

|AB - CD| + |AD - BC| \geq 2|AC - BD|.

1

An nxn Checkboard

Some checkers placed on an

n \times 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

(n^{2}-2)/3

checkers have been placed on the board.

1

usamo 99/4

a_{1}, a_{2}, \dots, a_{n}

n > 3

) be real numbers such that a_{1} + a_{2} + \cdots + a_{n} \geq n \qquad \mbox{and} \qquad a_{1}^{2} + a_{2}^{2} + \cdots + a_{n}^{2} \geq n^{2}. Prove that

\max(a_{1}, a_{2}, \dots, a_{n}) \geq 2