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Mock AIME 6 2006-2007 Problems/Problem 7

Revision as of 19:47, 26 November 2023 by Tomasdiaz (talk | contribs)

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 $x=e^{\frac{k}{n+1}2\pi i}$ for $k=1,2,\cdots,n$ with the only real solution when $n$ is odd and $k=\frac{n+1}{2}$ and the rest are complex.

Therefore, each $P_n(x)$ will have $n$ distinct complex roots when $n$ is even and $n-1$ distinct complex roots when $n$ is odd.

The roots of $Q_n(x)$ will be all of the roots of $P_1,P_2,\cdots, P_n$ which will include several repeated roots.

To get how many more complex roots does $Q_{1004}$ have than $Q_{1003}$ that will be the number of complex roots of $P_{1004}$.

But to get how many more distinct complex roots, we must subtract the complex roots of $P_{1004}$ that can be found in $Q_{1003}$

complex roots of $P_{1004}:\; x=e^{\frac{1}{1005}2\pi i},e^{\frac{2}{1005}2\pi i},e^{\frac{3}{1005}2\pi i},\cdots,e^{\frac{1004}{1005}2\pi i}$ for a total of $\textbf{1004}$ complex roots.

Now we subtract the common complex roots of $P_{1004}$ with $Q_{1003}$ by finding how many reducible fractions are there in $\frac{k}{1005}$ for $k=1,2,\cdots,1005$

Since $1005=(3)(5)(67)$ then $\frac{k}{1005}$ is reducible when $k \equiv 0\;(mod\;3)$ or $k \equiv 0\;(mod\;5)$ or $k \equiv 0\;(mod\;67)$

$k \equiv 0\;(mod\;3):$

$3, 6, 9, 12, 15,\cdots, 1002$ gives 334 terms

$k \equiv 0\;(mod\;5):$

$5, 10, 15, 20, 25,\cdots, 1000$ gives 200 terms

But we subtract the repeated terms in the 5 sequence that are also a multiple of 3:

$15, 30,\cdots,990$ which is 66 terms

$k \equiv 0\;(mod\;67):$

$67, 134, 201,\cdots, 928$ gives 14 terms

But we subtract the repeated terms in the 67 sequence that are also a multiple of 3 or 5:

$201, 335, 402, 603, 670, and 804$ gives 6 terms.

Total terms to subtract: $334+200-66+14-6=\textbf{476}$

Therefore, the number of distinct complex that $Q_{1004}$ have more than $Q_{1003}$ is:




~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.

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