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A sequence 1,2,2,3,3,3,4,4,4,4,5,5,5,5,5

Source: Saint Petersburg olympiad 2024, 11.2

September 22, 2024

algebrasequenses

Problem Statement

Given a sequence

a_n

:

1, 2, 2, 3, 3, 3, 4, 4, 4, 4, \dots

(one '1', two '2' and so on) and another sequence

b_n

such that

a_{b_n}=b_{a_n}

for all positive integers

n

. It is known that

b_k=1

for some

k>100

. Prove that

b_m=1

for all

m>k

.

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