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kn + k + n -> 9999999 or 99999999 or 48999999

Source: (2022 -) 2023 XVI Dürer Math Competition Finals Day 1 E2

May 25, 2024
number theory

Problem Statement

When Andris entered the room, there were the numbers 33 and 2424 on the board. In one step, if there are the (not necessarily different) numbers kk and nn on the board already, then Andris can write the numberkn+k+n kn + k + n on the board, too.
a) Can Andris write the number 99999999999999 on the board after a few moves? b) What if he wants to get 9999999999999999? c) And what about 4899999948999999?
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