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

(Created page with "==Problem== Let <math>a,b,c,d</math> be real numbers such that <math>b-d \ge 5</math> and all zeros <math>x_1, x_2, x_3,</math> and <math>x_4</math> of the polynomial <math>P(x)=...") |
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==Solution== | ==Solution== | ||

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

+ | <cmath> P(i) P(-i) = \left[ (b-d-1) + (a-c)i \right][ (b-d-1) - (a-c)i \right] = (b-d-1)^2 + (a-c)^2 \ge (5-1)^2 + 0^2 = 16 </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. |

## Revision as of 06:17, 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

\[P(i) P(-i) = \left[ (b-d-1) + (a-c)i \right][ (b-d-1) - (a-c)i \right] = (b-d-1)^2 + (a-c)^2 \ge (5-1)^2 + 0^2 = 16\] (Error compiling LaTeX. ! Extra \right.)

where . Equality holds if , so this bound is sharp.