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Romanian National Olympiad 2024 - Grade 9 - Problem 4

Source: Romanian National Olympiad 2024 - Grade 9 - Problem 4

April 6, 2024

algebraArithmetic ProgressionSequence

Problem Statement

Let

a

be a given positive integer. We consider the sequence

(x_n)_{n \ge 1}

defined by

x_n=\frac{1}{1+na},

for every positive integer

n.

Prove that for any integer

k \ge 3,

there exist positive integers

n_1<n_2<\ldots<n_k

such that the numbers

x_{n_1},x_{n_2},\ldots,x_{n_k}

are consecutive terms in an arithmetic progression.