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\lvert (b-d-1) + (a-c)i \right\rvert= (b-d-1)^2 + (a-c)^2 \ge16 </cmath> | + | <cmath> P(i) P(-i) = \left\lvert (b-d-1) + (a-c)i \right\rvert^2= (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 17:04, 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.