# AoPS Wiki talk:Problem of the Day/June 21, 2011

## Contents

## Problem

AoPSWiki:Problem of the Day/June 21, 2011

## Solutions

### First Solution

. Hence or . If and , then attains this maximum value on the circle .

### Second Solution

Let and be real numbers such that . Note that thus, we may assume that and are positive. Furthermore, by the Cauchy-Schwarz Inequality, we have but since , the inequality is equivalent with or so the maximum is and it is reached when .

### Third Solution

Imagine the equations graphed in the coordinate plane. is a circle centered at the origin with

radius . is a line. We want to find the largest value of such that the line

intersects the circle, giving real number solutions for and . This occurs when is tangent

to the circle, and thus when the distance from the line to the origin is . The distance from a point to a line is

.

Plugging in , and and setting the expression equal to yields

, or . We want the largest value of , so

is the highest possible value.

### Fourth Solution

By AM-QM, , so , equality when .