MathDB

6

Part of 2021 Regional Olympiad of Mexico Center Zone

Problems(1)

Multiplicativity+infinity=identity

Source: 2021 Mexico Center Zone Regional Olympiad, problem 6

1/17/2022
The sequence a1,a2,…a_1,a_2,\dots of positive integers obeys the following two conditions:
[*] For all positive integers m,nm,n, it happens that am⋅an=amna_m\cdot a_n=a_{mn} [*] There exist infinite positive integers nn such that (a1,a2,…,an)(a_1,a_2,\dots,a_n) is a permutation of (1,2,…,n)(1,2,\dots,n)
Prove that an=na_n=n for all positive integers nn.
Proposed by José Alejandro Reyes González
Mexicoalgebranumber theoryfunctional equationSequence