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The numbers $1, 2, 3,\ldots, 2\underbrace{00\ldots0}_{100 \text{ zeroes}}2$ are

Source: Caucasus MO 2023

July 16, 2023

Problem Statement

The numbers 1,2,3,…,200…0⏟100 zeroes21, 2, 3,\ldots, 2\underbrace{00\ldots0}_{100 \text{ zeroes}}2 are written on the board. Is it possible to paint half of them red and remaining ones blue, so that the sum of red numbers is divisible by the sum of blue ones?
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