15

Part of 2024 China Team Selection Test

Problems(1)

Fractional part of exponent of solution to polynomial

Source: 2024 CTST P15

n>1

is an integer. Let real number

x>1

x^{101}-nx^{100}+nx-1=0.

Prove that for any real

0<a<b<1

, there exists a positive integer

m

a<\{x^m\}<b.

Proposed by Chenjie Yu

algebrapolynomial2024 CTST