2014 USAMO Problems/Problem 1

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.

Hint

Factor $x^2 + 1$ as the product of two linear binomials.

Solution

Using the hint we turn the equation into $\prod_{k=1} ^4 (x_k-i)(x_k+i) \implies P(i)P(-i) \implies ((b-d-1)-i(a-c))^2 \implies \boxed{16}$ .

Solution

The value in question is equal to $P(i) P(-i) = \left\lvert (b-d-1) + (a-c)i \right\rvert^2= (b-d-1)^2 + (a-c)^2 \ge16$ 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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2014 USAMO Problems/Problem 1

Contents

Problem

Hint

Solution

Solution