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Positive integers partitioned into 2 sequences

Source: 2023 Serbia TST Problem 3

May 22, 2023

SerbiaTSTalgebrabroken

Problem Statement

The positive integers are partitioned into 2 sequences

a_1<a_2<\dots

and

b_1<b_2<\dots

such that

b_n=a_n+n

for every positive integer

n

.
Show that

a_n+b_n=a_{b_n}