Difference between revisions of "2014 USAMO Problems/Problem 1"
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− | Using the hint we turn the equation into <math>\prod_{k=1} ^4 (x_k-i)(x_k+i) \implies P(i)P(-i) \implies | + | Using the hint we turn the equation into <math>\prod_{k=1} ^4 (x_k-i)(x_k+i) \implies P(i)P(-i) \implies (b-d-1)^2 + (a-c)^2 \implies \boxed{16}</math>. This minimum is achieved when all the <math>x_i</math> are equal to <math>1</math>. |
Latest revision as of 02:14, 21 June 2019
Problem
Let be real numbers such that and all zeros and of the polynomial are real. Find the smallest value the product can take.
Hint
Factor as the product of two linear binomials.
Solution
Using the hint we turn the equation into . This minimum is achieved when all the are equal to .