2014 USAMO Problems/Problem 1

Revision as of 06:17, 30 April 2014 by V Enhance (talk | contribs) (Solution)

Problem

Let $a,b,c,d$ be real numbers such that $b-d \ge 5$ and all zeros $x_1, x_2, x_3,$ and $x_4$ of the polynomial $P(x)=x^4+ax^3+bx^2+cx+d$ are real. Find the smallest value the product $(x_1^2+1)(x_2^2+1)(x_3^2+1)(x_4^2+1)$ can take.

Solution

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 $i = \sqrt{-1}$. Equality holds if $x_1 = x_2 = x_3 = x_4 = 1$, so this bound is sharp.

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