# Difference between revisions of "2001 IMO Shortlist Problems/N2"

(New page: == Problem == Consider the system <math>x + y = z + u,</math> <math>2xy & = zu.</math> Find the greatest value of the real constant <math>m</math> such that <math>m \leq x/y</math> for any...) |
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Consider the system <math>x + y = z + u,</math> <math>2xy & = zu.</math> Find the greatest value of the real constant <math>m</math> such that <math>m \leq x/y</math> for any positive integer solution <math>(x,y,z,u)</math> of the system, with <math>x \geq y</math>. | Consider the system <math>x + y = z + u,</math> <math>2xy & = zu.</math> Find the greatest value of the real constant <math>m</math> such that <math>m \leq x/y</math> for any positive integer solution <math>(x,y,z,u)</math> of the system, with <math>x \geq y</math>. | ||

− | == Solution == | + | ==Solution== |

− | {{solution} | + | First consider the real solutions to the system. We have by AM-GM that <math>\frac{z+u}{2}\ge\sqrt{zu}</math> and substituting we get <math>\frac{x+y}{2}\ge\sqrt{2xy}</math>. Squaring and simplifying and dividing by <math>y^2</math>, we get the inequality <math>r^2-6r+1\ge0</math>, where <math>r=\frac{x}{y}</math>. Then <math>r^2-6r+9\ge8</math>, so <math>r\ge3+2\sqrt2</math> or <math>r\le3-2\sqrt2</math>. Since <math>r\ge1</math>, we discard the second inequality and have that <math>3+2\sqrt2</math> is a lower bound for <math>r</math> |

+ | |||

+ | This bound is also attainable for real values when <math>z=u</math>. Since <math>\mathbb{Q+}</math> is dense, it is always possible to assign rational values to <math>x, y, z,</math> and <math>w</math> so that <math>r</math> approaches <math>\sqrt3</math>, though equality is never reached. From any rational solution, it is possible to create an integer solution by multiplying by the least common multiple of the denominators and keep the same value of <math>r</math>. Thus, <math>\boxed{m=3+2\sqrt2}</math>. | ||

== Resources == | == Resources == |

## Revision as of 22:47, 24 April 2012

## Problem

Consider the system $2xy & = zu.$ (Error compiling LaTeX. ! Misplaced alignment tab character &.) Find the greatest value of the real constant such that for any positive integer solution of the system, with .

## Solution

First consider the real solutions to the system. We have by AM-GM that and substituting we get . Squaring and simplifying and dividing by , we get the inequality , where . Then , so or . Since , we discard the second inequality and have that is a lower bound for

This bound is also attainable for real values when . Since is dense, it is always possible to assign rational values to and so that approaches , though equality is never reached. From any rational solution, it is possible to create an integer solution by multiplying by the least common multiple of the denominators and keep the same value of . Thus, .