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Omni-MATH / Let p(x) be the polynomial (1−x)^(a)(1−x²)^(b)(1−x³)^(c)⋯(1−x³²)^(k) , where a,b,⋯,k are integers. When expanded in powers of x , the coefficient of x¹ is −2 and the coefficients o…

Problem

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problem

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

-2

and the coefficients of

x^2

x^3

, ...,

x^{32}

are all zero. Find

k

Plain-text mathematical notation (without MathML)

Let p(x) be the polynomial (1−x)^(a)(1−x²)^(b)(1−x³)^(c)⋯(1−x³²)^(k) , where a,b,⋯,k are integers. When expanded in powers of x , the coefficient of x¹ is −2 and the coefficients of x² , x³ , ..., x³² are all zero. Find k .

Original LaTeX notation

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 $k$ .

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task_id: adf4de60-2409-5684-ac87-4e67f87a1e05
task_key: test--adf4de60-2409-5684-ac87-4e67f87a1e05
task_revision_id: 4
config / split: / test

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