USA TST 2014 #4
by Wolstenholme, Oct 15, 2014, 8:59 PM
Let 
be a positive even integer, and let 
be real numbers satisfying ![\[ \sum_{i=1}^{n-1} \left\lvert c_i-1 \right\rvert < 1. \]](//latex.artofproblemsolving.com/b/2/4/b24ce35ea28b3898e6d2c4881752a1a9cc90087c.png)
Prove that ![\[
2x^n - c_{n-1}x^{n-1} + c_{n-2}x^{n-2} - \dots - c_1x^1 + 2.
\]](//latex.artofproblemsolving.com/d/3/7/d37aec11e6643012eeb44625189a619239fa1d05.png)
has no real roots.
Proof:
Let
for all integers 
. Thus the condition becomes 
and our polynomial can be written as 
(when 
). Now we proceed with some casework:
Case 1:
Here we have
as desired.
Case 2:
or 
Here we have
as desired.
Case 3:
Here we have that
as desired.
Therefore
for all 
which implies the desired result.
be a positive even integer, and let 
be real numbers satisfying ![\[ \sum_{i=1}^{n-1} \left\lvert c_i-1 \right\rvert < 1. \]](http://latex.artofproblemsolving.com/b/2/4/b24ce35ea28b3898e6d2c4881752a1a9cc90087c.png)
Prove that ![\[
2x^n - c_{n-1}x^{n-1} + c_{n-2}x^{n-2} - \dots - c_1x^1 + 2.
\]](http://latex.artofproblemsolving.com/d/3/7/d37aec11e6643012eeb44625189a619239fa1d05.png)
has no real roots.Proof:
Let
for all integers 
. Thus the condition becomes 
and our polynomial can be written as 
(when 
). Now we proceed with some casework:Case 1:
Here we have
as desired.Case 2:
or 
Here we have
as desired.Case 3:
Here we have that
as desired.Therefore
for all 
which implies the desired result.
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