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Equal number of 1s, 2s, & 3s in a circular sequence

Source: XVII Tuymaada Mathematical Olympiad (2010), Senior Level

July 31, 2011
combinatorics unsolvedcombinatorics

Problem Statement

Arranged in a circle are 20102010 digits, each of them equal to 11, 22, or 33. For each positive integer kk, it's known that in any block of 3k3k consecutive digits, each of the digits appears at most k+10k+10 times. Prove that there is a block of several consecutive digits with the same number of 11s, 22s, and 33s.
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