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Putnam 1985 B1

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August 5, 2019
Putnamalgebrapolynomial

Problem Statement

Let kk be the smallest positive integer for which there exist distinct integers m1,m2,m3,m4,m5m_{1}, m_{2}, m_{3}, m_{4}, m_{5} such that the polynomial p(x)=(x−m1)(x−m2)(x−m3)(x−m4)(x−m5)p(x)=\left(x-m_{1}\right)\left(x-m_{2}\right)\left(x-m_{3}\right)\left(x-m_{4}\right)\left(x-m_{5}\right) has exactly kk nonzero coefficients. Find, with proof, a set of integers m1,m2,m3,m4,m5m_{1}, m_{2}, m_{3}, m_{4}, m_{5} for which this minimum kk is achieved.
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