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