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a_n/g_n is a positive integer, then x_1 = x_2 = ··· = x_n

Source: 1975 German Federal - Bundeswettbewerb Mathematik - BWM - Round 2 p3

November 20, 2022
number theoryarithmetic meangeometric mea

Problem Statement

For nnpositive integers x1,x2,...,xn x_1,x2,...,x_n, ana_n is their arithmetic and gng_n the geometric mean. Consider the statement SnS_n: If an/gna_n/g_n is a positive integer, then x1=x2=⋅⋅⋅=xnx_1 = x_2 = ··· = x_n. Prove S2S_2 and disprove SnS_n for all even n>2n > 2.
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