Difference between revisions of "2014 USAMO Problems/Problem 1"
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Factor <math>x^2 + 1</math> as the product of two linear binomials. | 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)-(a-c) | + | 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)-i(a-c))^2 \implies \boxed{16}</math>. |
==Solution== | ==Solution== |
Revision as of 16:50, 23 November 2016
Contents
[hide]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 .
Solution
The value in question is equal to
where
. Equality holds if
, so this bound is sharp.