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Fractional part of exponent of solution to polynomial

Source: 2024 CTST P15

March 24, 2024
algebrapolynomial2024 CTST

Problem Statement

n>1n>1 is an integer. Let real number x>1x>1 satisfy x101−nx100+nx−1=0.x^{101}-nx^{100}+nx-1=0. Prove that for any real 0<a<b<10<a<b<1, there exists a positive integer mm so that a<{xm}<b.a<\{x^m\}<b.
Proposed by Chenjie Yu
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