# Difference between revisions of "2015 USAMO Problems"

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===Problem 6=== | ===Problem 6=== | ||

Consider <math>0<\lambda<1</math>, and let <math>A</math> be a multiset of positive integers. Let <math>A_n=\{a\in A: a\leq n\}</math>. Assume that for every <math>n\in\mathbb{N}</math>, the set <math>A_n</math> contains at most <math>n\lambda</math>numbers. Show that there are infinitely many <math>n\in\mathbb{N}</math> for which the sum of the elements in <math>A_n</math> is at most <math>\frac{n(n+1)}{2}\lambda</math>. (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 <math>\{1, 2, 3\}</math> and <math>\{2, 1, 3\}</math> are equivalent, but <math>\{1, 1, 2, 3\}</math> and <math>\{1, 2, 3\}</math> differ.) | Consider <math>0<\lambda<1</math>, and let <math>A</math> be a multiset of positive integers. Let <math>A_n=\{a\in A: a\leq n\}</math>. Assume that for every <math>n\in\mathbb{N}</math>, the set <math>A_n</math> contains at most <math>n\lambda</math>numbers. Show that there are infinitely many <math>n\in\mathbb{N}</math> for which the sum of the elements in <math>A_n</math> is at most <math>\frac{n(n+1)}{2}\lambda</math>. (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 <math>\{1, 2, 3\}</math> and <math>\{2, 1, 3\}</math> are equivalent, but <math>\{1, 1, 2, 3\}</math> and <math>\{1, 2, 3\}</math> differ.) | ||

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## Revision as of 09:36, 4 February 2016

## Contents

## Day 1

### Problem 1

Solve in integers the equation

### Problem 2

Quadrilateral is inscribed in circle with and . Let be a variable point on segment . Line meets again at (other than ). Point lies on arc of such that is perpendicular to . Let denote the midpoint of chord . As varies on segment , show that moves along a circle.

### Problem 3

Let , where . Each of the subsets of is to be colored red or blue. (The subset itself is assigned a color and not its individual elements.) For any set , we then write for the number of subsets of T that are blue.

Determine the number of colorings that satisfy the following condition: for any subsets and of ,

## Day 2

### Problem 4

Steve is piling indistinguishable stones on the squares of an grid. Each square can have an arbitrarily high pile of stones. After he finished piling his stones in some manner, he can then perform stone moves, defined as follows. Consider any four grid squares, which are corners of a rectangle, i.e. in positions for some , such that and . A stone move consists of either removing one stone from each of and and moving them to and respectively,j or removing one stone from each of and and moving them to and respectively.

Two ways of piling the stones are equivalent if they can be obtained from one another by a sequence of stone moves.

How many different non-equivalent ways can Steve pile the stones on the grid?

### Problem 5

Let be distinct positive integers such that . Show that is a composite number.

### Problem 6

Consider , and let be a multiset of positive integers. Let . Assume that for every , the set contains at most numbers. Show that there are infinitely many for which the sum of the elements in is at most . (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 and are equivalent, but and differ.)

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.