2000 USAMO

Part of USAMO

Subcontests

(6)

1

USAMO 2000 Problem 5

A_1A_2A_3

be a triangle and let

\omega_1

be a circle in its plane passing through

A_1

A_2.

Suppose there exist circles

\omega_2, \omega_3, \dots, \omega_7

k = 2, 3, \dots, 7,

\omega_k

is externally tangent to

\omega_{k-1}

and passes through

A_k

A_{k+1},

A_{n+3} = A_{n}

n \ge 1

\omega_7 = \omega_1.

1

USAMO 2000 Problem 4

Find the smallest positive integer

n

n

1000 \times 1000

chessboard are colored, then there will exist three colored squares whose centers form a right triangle with sides parallel to the edges of the board.

1

USAMO 2000 Problem 3

A game of solitaire is played with

R

W

white cards, and

B

blue cards. A player plays all the cards one at a time. With each play he accumulates a penalty. If he plays a blue card, then he is charged a penalty which is the number of white cards still in his hand. If he plays a white card, then he is charged a penalty which is twice the number of red cards still in his hand. If he plays a red card, then he is charged a penalty which is three times the number of blue cards still in his hand. Find, as a function of

R, W,

B,

the minimal total penalty a player can amass and all the ways in which this minimum can be achieved.

1

USAMO 2000 Problem 2

S

be the set of all triangles

ABC

5 \left( \dfrac{1}{AP} + \dfrac{1}{BQ} + \dfrac{1}{CR} \right) - \dfrac{3}{\min\{ AP, BQ, CR \}} = \dfrac{6}{r},

r

is the inradius and

P, Q, R

are the points of tangency of the incircle with sides

AB, BC, CA,

respectively. Prove that all triangles in

S

are isosceles and similar to one another.

1

Very concex function

Call a real-valued function

f

very convex if \frac {f(x) \plus{} f(y)}{2} \ge f\left(\frac {x \plus{} y}{2}\right) \plus{} |x \minus{} y| holds for all real numbers

x

y

. Prove that no very convex function exists.

1

The hardest USAMO inequality problem

a_1, b_1, a_2, b_2, \dots , a_n, b_n

be nonnegative real numbers. Prove that

\sum_{i, j = 1}^{n} \min\{a_ia_j, b_ib_j\} \le \sum_{i, j = 1}^{n} \min\{a_ib_j, a_jb_i\}.