Difference between revisions of "2014 USAMO Problems/Problem 1"
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Revision as of 06:17, 30 April 2014
Let be real numbers such that and all zeros and of the polynomial are real. Find the smallest value the product can take.
The value in question is equal to
\[P(i) P(-i) = \left[ (b-d-1) + (a-c)i \right][ (b-d-1) - (a-c)i \right] = (b-d-1)^2 + (a-c)^2 \ge (5-1)^2 + 0^2 = 16\] (Error compiling LaTeX. ! Extra \right.)
where . Equality holds if , so this bound is sharp.