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Even Algebra Can Be Cute

Source: Ukrainian Mathematical Olympiad 2023. Day 1, Problem 10.4

April 5, 2023

algebraSequences

Problem Statement

Let

(x_n)

be an infinite sequence of real numbers from interval

(0, 1)

. An infinite sequence

(a_n)

of positive integers is defined as follows:

a_1 = 1

, and for

i \ge 1

a_{i+1}

is equal to the smallest positive integer

m

, for which

[x_1 + x_2 + \ldots + x_m] = a_i

. Show that for any indexes

i, j

holds

a_{i+j} \ge a_i + a_j

.
Proposed by Nazar Serdyuk