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. | ||
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==Solution== | ==Solution== | ||
Revision as of 20:04, 14 May 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.
Hint
Factor 
as the product of two linear binomials.
Solution
The value in question is equal to
![\[P(i) P(-i) = \left\lvert (b-d-1) + (a-c)i \right\rvert^2= (b-d-1)^2 + (a-c)^2 \ge16\]](http://latex.artofproblemsolving.com/6/8/8/68878968cd413cfea0ee927077f60df9fdfd792f.png)
where 
. Equality holds if 
, so this bound is sharp.
