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

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==Problem== | ==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)=x^4+ax^3+bx^2+cx+d</math> are real. Find the smallest value the product <math>(x_1^2+1)(x_2^2+1)(x_3^2+1)(x_4^2+1)</math> can take. | 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)=x^4+ax^3+bx^2+cx+d</math> are real. Find the smallest value the product <math>(x_1^2+1)(x_2^2+1)(x_3^2+1)(x_4^2+1)</math> can take. | ||

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+ | ==Hint== | ||

+ | Factor <math>x^2 + 1</math> as the product of two linear binomials. | ||

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

− | + | Using the hint we turn the equation into <math>\prod_{k=1} ^4 (x_k-i)(x_k+i) \implies P(i)P(-i) \implies (b-d-1)^2 + (a-c)^2 \implies \boxed{16}</math>. This minimum is achieved when all the <math>x_i</math> are equal to <math>1</math>. | |

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## Latest revision as of 02:14, 21 June 2019

## Problem

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

## Hint

Factor as the product of two linear binomials.

## Solution

Using the hint we turn the equation into . This minimum is achieved when all the are equal to .