2002 USAMO

Part of USAMO

Subcontests

(5)

1

USAMO 2002 Problem 5

a,b

be integers greater than 2. Prove that there exists a positive integer

k

and a finite sequence

n_1, n_2, \dots, n_k

of positive integers such that

n_1 = a

n_k = b

n_i n_{i+1}

is divisible by

n_i + n_{i+1}

i

1 \leq i < k

1

USAMO 2002 Problem 4

\mathbb{R}

be the set of real numbers. Determine all functions

f: \mathbb{R} \to \mathbb{R}

f(x^2 - y^2) = x f(x) - y f(y)

for all pairs of real numbers

x

y

1

USAMO 2002 Problem 6

n \times n

sheet of stamps, from which I've been asked to tear out blocks of three adjacent stamps in a single row or column. (I can only tear along the perforations separating adjacent stamps, and each block must come out of the sheet in one piece.) Let

b(n)

be the smallest number of blocks I can tear out and make it impossible to tear out any more blocks. Prove that there are real constants

c

d

\dfrac{1}{7} n^2 - cn \leq b(n) \leq \dfrac{1}{5} n^2 + dn

n > 0

1

USAMO 2002 Problem 3

Prove that any monic polynomial (a polynomial with leading coefficient 1) of degree

n

with real coefficients is the average of two monic polynomials of degree

n

n

1

USAMO 2002 Problem 2

ABC

be a triangle such that

\left( \cot \dfrac{A}{2} \right)^2 + \left( 2\cot \dfrac{B}{2} \right)^2 + \left( 3\cot \dfrac{C}{2} \right)^2 = \left( \dfrac{6s}{7r} \right)^2,

s

r

denote its semiperimeter and its inradius, respectively. Prove that triangle

ABC

is similar to a triangle

T

whose side lengths are all positive integers with no common divisors and determine these integers.