Difference between revisions of "2014 USAMO Problems/Problem 1"
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==Solution== | ==Solution== | ||
The value in question is equal to | The value in question is equal to | ||
− | <cmath> P(i) P(-i) = \left | + | <cmath> P(i) P(-i) = \left\lvert (b-d-1) + (a-c)i \right\rvert= (b-d-1)^2 + (a-c)^2 \ge16 </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. | 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 05:18, 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
where
. Equality holds if
, so this bound is sharp.