# Difference between revisions of "2014 USAMO Problems"

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[[2014 USAMO Problems/Problem 4|Solution]] | [[2014 USAMO Problems/Problem 4|Solution]] | ||

===Problem 5=== | ===Problem 5=== | ||

+ | Let <math>ABC</math> be a triangle with orthocenter <math>H</math> and let <math>P</math> be the second intersection of the circumcircle of triangle <math>AHC</math> with the internal bisector of the angle <math>\angle BAC</math>. Let <math>X</math> be the circumcenter of triangle <math>AHC</math> and <math>Y</math> the orthocenter of triangle <math>APC</math>. Prove that the length of segment <math>XY</math> is equal to the circumradius of triangle <math>ABC</math>. | ||

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[[2014 USAMO Problems/Problem 5|Solution]] | [[2014 USAMO Problems/Problem 5|Solution]] | ||

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

Prove that there is a constant <math>c>0</math> with the following property: If <math>a, b, n</math> are positive integers such that <math>\gcd(a+i, b+j)>1</math> for all <math>i, j\in\{0, 1, \ldots n\}</math>, then<cmath>\min\{a, b\}>c^n\cdot n^{\frac{n}{2}}.</cmath> | Prove that there is a constant <math>c>0</math> with the following property: If <math>a, b, n</math> are positive integers such that <math>\gcd(a+i, b+j)>1</math> for all <math>i, j\in\{0, 1, \ldots n\}</math>, then<cmath>\min\{a, b\}>c^n\cdot n^{\frac{n}{2}}.</cmath> | ||

[[2014 USAMO Problems/Problem 6|Solution]] | [[2014 USAMO Problems/Problem 6|Solution]] |

## Revision as of 16:46, 30 April 2014

## Contents

## Day 1

### Problem 1

Let be real numbers such that and all zeros and of the polynomial are real. Find the smallest value the product can take.

### Problem 2

Let be the set of integers. Find all functions such that for all with .

### Problem 3

Prove that there exists an infinite set of points in the plane with the following property: For any three distinct integers and , points , , and are collinear if and only if .

## Day 2

### Problem 4

Let be a positive integer. Two players and play a game on an infinite grid of regular hexagons. Initially all the grid cells are empty. Then the players alternately take turns with moving first. In his move, may choose two adjacent hexagons in the grid which are empty and place a counter in both of them. In his move, may choose any counter on the board and remove it. If at any time there are consecutive grid cells in a line all of which contain a counter, wins. Find the minimum value of for which cannot win in a finite number of moves, or prove that no such minimum value exists.

### Problem 5

Let be a triangle with orthocenter and let be the second intersection of the circumcircle of triangle with the internal bisector of the angle . Let be the circumcenter of triangle and the orthocenter of triangle . Prove that the length of segment is equal to the circumradius of triangle .

### Problem 6

Prove that there is a constant with the following property: If are positive integers such that for all , then