Difference between revisions of "2014 USAMO Problems/Problem 1"
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==Hint== | ==Hint== | ||
Factor <math>x^2 + 1</math> as the product of two linear binomials. | Factor <math>x^2 + 1</math> as the product of two linear binomials. | ||
| − | + | ==Solution== | |
| − | + | 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)-(a-c) +1)^2 \implies \boxed{16}</math>. | |
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==Solution== | ==Solution== | ||
Revision as of 16:49, 23 November 2016
Contents
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 
.
Solution
The value in question is equal to
![\[P(i) P(-i) = \left\lvert (b-d-1) + (a-c)i \right\rvert^2= (b-d-1)^2 + (a-c)^2 \ge16\]](http://latex.artofproblemsolving.com/6/8/8/68878968cd413cfea0ee927077f60df9fdfd792f.png)
where 
. Equality holds if 
, so this bound is sharp.
