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

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

The value in question is equal to | The value in question is equal to | ||

− | <cmath> P(i) P(-i) = \left | + | <cmath> P(i) P(-i) = \left\lvert (b-d-1) + (a-c)i \right\rvert= (b-d-1)^2 + (a-c)^2 \ge16 </cmath> |

where <math>i = \sqrt{-1}</math>. Equality holds if <math>x_1 = x_2 = x_3 = x_4 = 1</math>, so this bound is sharp. | where <math>i = \sqrt{-1}</math>. Equality holds if <math>x_1 = x_2 = x_3 = x_4 = 1</math>, so this bound is sharp. |

## Revision as of 06:18, 30 April 2014

## Problem

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

## Solution

The value in question is equal to where . Equality holds if , so this bound is sharp.