1985 Putnam B1

Let $k$ be the smallest positive integer for which there exist

distinct integers $m_1, m_2, m_3, m_4, m_5$ such that the polynomial

$$

p(x) = (x-m_1)(x-m_2)(x-m_3)(x-m_4)(x-m_5)

$$

has exactly $k$ nonzero coefficients. Find, with proof, a set of

integers $m_1, m_2, m_3, m_4, m_5$ for which this minimum $k$ is achieved.

令 $k$ 为存在的最小正整数

不同的整数 $m_1, m_2, m_3, m_4, m_5$ 使得多项式

$$

p(x) = (x-m_1)(x-m_2)(x-m_3)(x-m_4)(x-m_5)

$$

恰好有 $k$ 个非零系数。找到一组有证据的

达到最小 $k$ 的整数 $m_1、m_2、m_3、m_4、m_5$。