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>. |
Revision as of 01:14, 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
.