2014 USAMO Problems/Problem 1
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.