2012 USAMO

Part of USAMO

Subcontests

(4)

1

Bound on subsets

n\geq2

x_1, x_2, \ldots, x_n

be real numbers satisfying

x_1+x_2+\ldots+x_n=0, \qquad \text{and}\qquad x_1^2+x_2^2+\ldots+x_n^2=1.

For each subset

A\subseteq\{1, 2, \ldots, n\}

S_A=\sum_{i\in A}x_i.

A

is the empty set, then

S_A=0

.)
Prove that for any positive number

\lambda

, the number of sets

A

S_A\geq\lambda

2^{n-3}/\lambda^2

. For which choices of

x_1, x_2, \ldots, x_n, \lambda

does equality hold?

1

Functional Equation

Find all functions

f:\mathbb{Z}^+ \rightarrow \mathbb{Z}^+

\mathbb{Z}^+

is the set of positive integers) such that

f(n!) = f(n)!

for all positive integers

n

m-n

f(m) - f(n)

for all distinct positive integers

m, n

1

Sequence of integers

Determine which integers

n > 1

have the property that there exists an infinite sequence

a_1, a_2, a_3, \ldots

of nonzero integers such that the equality

a_k+2a_{2k}+\ldots+na_{nk}=0

holds for every positive integer

k

1

Circles colored with arcs

A circle is divided into

432

congruent arcs by

432

points. The points are colored in four colors such that some

108

points are colored Red, some

108

points are colored Green, some

108

points are colored Blue, and the remaining

108

points are colored Yellow. Prove that one can choose three points of each color in such a way that the four triangles formed by the chosen points of the same color are congruent.