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

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==Problem== | ==Problem== | ||

− | Let <math>a</math>, <math>b</math>, <math>c</math> be real numbers greater than or equal to <math>1</math>. Prove that <cmath>\min{\left (\frac{10a^2-5a+1}{b^2-5b+ | + | Let <math>a</math>, <math>b</math>, <math>c</math> be real numbers greater than or equal to <math>1</math>. Prove that <cmath>\min{\left (\frac{10a^2-5a+1}{b^2-5b+10},\frac{10b^2-5b+1}{c^2-5c+10},\frac{10c^2-5c+1}{a^2-5a+10}\right )}\leq abc </cmath> |

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==Solution== | ==Solution== | ||

Since <math>(a-1)^5\geqslant 0</math>, | Since <math>(a-1)^5\geqslant 0</math>, |

## Revision as of 16:17, 23 July 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,