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MMO 216 Moscow MO 1952 integer sequence with sum of squares

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August 8, 2019
Integer sequenceSequenceSum of SquaresDigitsnumber theory

Problem Statement

A sequence of integers is constructed as follows: a1a_1 is an arbitrary three-digit number, a2a_2 is the sum of squares of the digits of a1,a3a_1, a_3 is the sum of squares of the digits of a2a_2, etc. Prove that either 11 or 44 must occur in the sequence a1,a2,a3,....a_1, a_2, a_3, ....
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