2001 USAMO

Part of USAMO

Subcontests

(6)

1

USAMO 2001 Problem 6

Each point in the plane is assigned a real number such that, for any triangle, the number at the center of its inscribed circle is equal to the arithmetic mean of the three numbers at its vertices. Prove that all points in the plane are assigned the same number.

1

USAMO 2001 Problem 5

S

be a set of integers (not necessarily positive) such that (a) there exist

a,b \in S

\gcd(a,b)=\gcd(a-2,b-2)=1

x

y

are elements of

S

(possibly equal), then

x^2-y

also belongs to

S

S

is the set of all integers.

1

USAMO 2001 Problem 4

P

be a point in the plane of triangle

ABC

such that the segments

PA

PB

PC

are the sides of an obtuse triangle. Assume that in this triangle the obtuse angle opposes the side congruent to

PA

\angle BAC

1

USAMO 2001 Problem 2

ABC

be a triangle and let

\omega

be its incircle. Denote by

D_1

E_1

the points where

\omega

is tangent to sides

BC

AC

, respectively. Denote by

D_2

E_2

the points on sides

BC

AC

, respectively, such that

CD_2=BD_1

CE_2=AE_1

, and denote by

P

the point of intersection of segments

AD_2

BE_2

\omega

intersects segment

AD_2

at two points, the closer of which to the vertex

A

Q

AQ=D_2P

1

USAMO 2001 Problem 1

Each of eight boxes contains six balls. Each ball has been colored with one of

n

colors, such that no two balls in the same box are the same color, and no two colors occur together in more than one box. Determine, with justification, the smallest integer

n

for which this is possible.

1

Inequality with a^2+b^2+c^2+abc=4

a, b, c \geq 0

a^2+b^2+c^2 +abc = 4 .

0 \le ab + bc + ca - abc \leq 2.