IMO 2023 Shortlist N6
A sequence of integers a0,a1,a2,... is called kawaii, if a0 “ 0,a1 “ 1, and, for any positive integer n, we have pan1 ´ ...
Category: Number Theory
Problem
A sequence of integers a0,a1,a2,... is called kawaii, if a0 “ 0,a1 “ 1, and, for any
positive integer n, we have
pan1 ´ 3an 2an´1qpan1 ´ 4an 3an´1q “ 0.
An integer is called kawaii if it belongs to a kawaii sequence.
Suppose that two consecutive positive integers m and m`1 are both kawaii (not necessarily
belonging to the same kawaii sequence). Prove that 3 divides m, and that m{3 is kawaii.
(China)