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Putnam 1986 A6

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August 5, 2019
Putnamalgebrapolynomial

Problem Statement

Let a1,a2,…,ana_1, a_2, \dots, a_n be real numbers, and let b1,b2,…,bnb_1, b_2, \dots, b_n be distinct positive integers. Suppose that there is a polynomial f(x)f(x) satisfying the identity (1−x)nf(x)=1+∑i=1naixbi. (1-x)^n f(x) = 1 + \sum_{i=1}^n a_i x^{b_i}. Find a simple expression (not involving any sums) for f(1)f(1) in terms of b1,b2,…,bnb_1, b_2, \dots, b_n and nn (but independent of a1,a2,…,ana_1, a_2, \dots, a_n).
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