2003 USAMO

Part of USAMO

Subcontests

(5)

1

USAMO 2003 Problem 6

At the vertices of a regular hexagon are written six nonnegative integers whose sum is

2003^{2003}

. Bert is allowed to make moves of the following form: he may pick a vertex and replace the number written there by the absolute value of the difference between the numbers written at the two neighboring vertices. Prove that Bert can make a sequence of moves, after which the number 0 appears at all six vertices.

1

USAMO 2003 Problem 5

a

b

c

be positive real numbers. Prove that \dfrac{(2a \plus{} b \plus{} c)^2}{2a^2 \plus{} (b \plus{} c)^2} \plus{} \dfrac{(2b \plus{} c \plus{} a)^2}{2b^2 \plus{} (c \plus{} a)^2} \plus{} \dfrac{(2c \plus{} a \plus{} b)^2}{2c^2 \plus{} (a \plus{} b)^2} \le 8.

1

USAMO 2003 Problem 4

ABC

be a triangle. A circle passing through

A

B

intersects segments

AC

BC

D

E

, respectively. Lines

AB

DE

F

BD

CF

M

MF = MC

MB\cdot MD = MC^2

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USAMO 2003 Problem 3

n \neq 0

. For every sequence of integers

A = a_0,a_1,a_2,\dots, a_n

0 \le a_i \le i

i=0,\dots,n

, define another sequence

t(A)= t(a_0), t(a_1), t(a_2), \dots, t(a_n)

t(a_i)

to be the number of terms in the sequence

A

that precede the term

a_i

and are different from

a_i

. Show that, starting from any sequence

A

as above, fewer than

n

applications of the transformation

t

lead to a sequence

B

t(B) = B

1

USAMO 2003 Problem 2

A convex polygon

\mathcal{P}

in the plane is dissected into smaller convex polygons by drawing all of its diagonals. The lengths of all sides and all diagonals of the polygon

\mathcal{P}

are rational numbers. Prove that the lengths of all sides of all polygons in the dissection are also rational numbers.