Difference between revisions of "Mock AIME 6 2006-2007 Problems/Problem 7"

Revision as of 19:09, 26 November 2023

Problem

Let $P_n(x)=1+x+x^2+\cdots+x^n$ and $Q_n(x)=P_1\cdot P_2\cdots P_n$ for all integers $n\ge 1$ . How many more distinct complex roots does $Q_{1004}$ have than $Q_{1003}$ ?

Solution

The roots of $P_n(x)$ will be in the form $x=e^{\frac{2\pi k}{n+1}}$ for $k=1,2,\cdots,n$ with the only real solution when $n$ is odd and $k=\frac{n+1}{2}$

~Tomas Diaz. orders@tomasdiaz.com

Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.

@@ Line 4: / Line 4: @@
 ==Solution==
-The solution of <math>P_n(x)</math> will be in the form <math>x=e^{\frac{2\pi k}{n+1}}</math> for <math>k=1,2,\cdots,n</math>
+The roots of <math>P_n(x)</math> will be in the form <math>x=e^{\frac{2\pi k}{n+1}}</math> for <math>k=1,2,\cdots,n</math> with the only real solution when <math>n</math> is odd and <math>k=\frac{n+1}{2}</math>
 ~Tomas Diaz. orders@tomasdiaz.com
 {{alternate solutions}}

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Difference between revisions of "Mock AIME 6 2006-2007 Problems/Problem 7"

Revision as of 19:09, 26 November 2023

Problem

Solution