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A process with boxes containing stones

Source: 2019 Belarus Team Selection Test 2.3

September 2, 2019
number theoryprimitive rootcombinatorics

Problem Statement

10191019 stones are placed into two non-empty boxes. Each second Alex chooses a box with an even amount of stones and shifts half of these stones into another box. Prove that for each kk, 1k10181\le k\le1018, at some moment there will be a box with exactly kk stones.
(O. Izhboldin)