2014 USAMO Problems/Problem 1: Difference between revisions
Created page with "==Problem== Let <math>a,b,c,d</math> be real numbers such that <math>b-d \ge 5</math> and all zeros <math>x_1, x_2, x_3,</math> and <math>x_4</math> of the polynomial <math>P(x)=..." |
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==Solution== | ==Solution== | ||
The value in question is equal to | |||
<cmath> 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 </cmath> | |||
where <math>i = \sqrt{-1}</math>. Equality holds if <math>x_1 = x_2 = x_3 = x_4 = 1</math>, so this bound is sharp. | |||
Revision as of 05:17, 30 April 2014
Problem
Let 
be real numbers such that 
and all zeros 
and 
of the polynomial 
are real. Find the smallest value the product 
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. Unknown error_msg)
where 
. Equality holds if 
, so this bound is sharp.
