The Perplexing Polynomial

Algebra Level pending

This is not an original 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, ..., k are integers. When expanded in powers of x, the coefficient of x is -2 and the coefficients of \({ x }^{ 2 },{ x }^{ 3 },...,{ x }^{ 32 }\) are all zero.

Find k. Then, write k in the form \( { 2 }^{ n }-2^{ m } \) What is n+m?

based on a problem from the USAMO

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