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HMMT 2018 A5

  • Algebra and Number Theory
  • Answer

Let ω1,ω2,…,ω100\omega_{1}, \omega_{2}, \ldots, \omega_{100} be the roots of x101−1x−1\frac{x^{101}-1}{x-1} (in some order). Consider the set

S={ω11,ω22,ω33,…,ω100100}S=\left\{\omega_{1}^{1}, \omega_{2}^{2}, \omega_{3}^{3}, \ldots, \omega_{100}^{100}\right\}

Let MM be the maximum possible number of unique values in SS, and let NN be the minimum possible number of unique values in SS. Find M−NM-N.

Dataset text NuminaMath olympiads. Quoted with source.

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