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m^{x_0} =\sum_{i=1}^{1001}m^{x_i}

Source: 2019 Dürer Math Competition Finals Day2 E+15 https://artofproblemsolving.com/community/c1621835_2019_

January 5, 2022

diophantineDiophantine equationnumber theorySum

Problem Statement

The positive integer

m

and non-negative integers

x_0, x_1,..., x_{1001}

satisfy the following equation:

m^{x_0} =\sum_{i=1}^{1001}m^{x_i}.

How many possibilities are there for the value of

m