2015 USAMO

Part of USAMO

Subcontests

(3)

1

Funny sequence

0<\lambda<1

A

be a multiset of positive integers. Let

A_n=\{a\in A: a\leq n\}

. Assume that for every

n\in\mathbb{N}

A_n

contains at most

n\lambda

numbers. Show that there are infinitely many

n\in\mathbb{N}

for which the sum of the elements in

A_n

\frac{n(n+1)}{2}\lambda

. (A multiset is a set-like collection of elements in which order is ignored, but repetition of elements is allowed and multiplicity of elements is significant. For example, multisets

\{1, 2, 3\}

\{2, 1, 3\}

are equivalent, but

\{1, 1, 2, 3\}

\{1, 2, 3\}

1

On a^4+b^4=c^4+d^4=e^5

a

b

c

d

e

be distinct positive integers such that

a^4+b^4=c^4+d^4=e^5

ac+bd

is a composite number.

1

Subset coloring

S = \left\{ 1,2,\dots,n \right\}

n \ge 1

2^n

S

is to be colored red or blue. (The subset itself is assigned a color and not its individual elements.) For any set

T \subseteq S

, we then write

f(T)

for the number of subsets of

T

that are blue.
Determine the number of colorings that satisfy the following condition: for any subsets

T_1

T_2

S

f(T_1)f(T_2) = f(T_1 \cup T_2)f(T_1 \cap T_2).