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Let $p(x)$ be the polynomial $(1-x)^a(1-x^2)^b(1-x^3)^c\cdots(1-x^{32})^k$, where $a, b, \cdots, k$ are integers. When expanded in powers of $x$, the coefficient of $x^1$ is $-2$ and the coefficients of $x^2$, $x^3$, ..., $x^{32}$ are all zero. Find $k$.

This is a question from the 1988 USAMO, Problem 5. I didn't understand the solution. Can anyone help?

 Jan 20, 2020
 #1
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I never answer questions when they are presented with unrendered LaTex.

 Jan 20, 2020
 #2
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\(p(x)\) be the polynomial

\((1-x)^a*(1-x^2)^b*(1-x^3)^c...(1-x^{32})^k\)

Where a,b,c....k are integers. When expanded in powers of x, the coefficient of \(x^1\) is \(-2\) and the coefficients of \(x^2,x^3,....x^{32}\)are all zero.

Find k

 Jan 20, 2020
 #3
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Link to the question and answer:  https://artofproblemsolving.com/wiki/index.php/1988_USAMO_Problems/Problem_5

Guest Jan 20, 2020

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