# Difference between revisions of "2014 USAJMO Problems/Problem 1"

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<cmath>10a^2-5a+1\leqslant a^3(a^2-5a+10)</cmath> | <cmath>10a^2-5a+1\leqslant a^3(a^2-5a+10)</cmath> | ||

− | Since <math>a^2-5a+10=\left( a-\dfrac{5}{2}\right) +\dfrac{15}{4}\geqslant 0</math>, | + | Since <math>a^2-5a+10=\left( a-\dfrac{5}{2}\right)^2 +\dfrac{15}{4}\geqslant 0</math>, |

<cmath> \frac{10a^2-5a+1}{a^2-5a+10}\leqslant a^3 </cmath> | <cmath> \frac{10a^2-5a+1}{a^2-5a+10}\leqslant a^3 </cmath> | ||

Also note that <math>10a^2-5a+1=10\left( a-\dfrac{1}{4}\right)+\dfrac{3}{8}\geqslant 0</math>, | Also note that <math>10a^2-5a+1=10\left( a-\dfrac{1}{4}\right)+\dfrac{3}{8}\geqslant 0</math>, |

## Revision as of 23:52, 29 April 2014

## Problem

Let , , be real numbers greater than or equal to . Prove that

## Solution

Since , or Since , Also note that , We conclude Similarly, So or Therefore,