MathDB

3

Part of 2018 Turkey MO (2nd Round)

Problems(1)

Integrality of a certain quantity

Source: Turkey National Mathematical Olympiad 2018

12/2/2018
A sequence a1,a2,…a_1,a_2,\dots satisfy ∑i=1na⌊ni⌋=n10, \sum_{i =1}^n a_{\lfloor \frac{n}{i}\rfloor }=n^{10}, for every n∈Nn\in\mathbb{N}. Let cc be a positive integer. Prove that, for every positive integer nn, can−can−1n \frac{c^{a_n}-c^{a_{n-1}}}{n} is an integer.
Sequencesalgebranumber theoryTurkey