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A well know problem about sum-of-digit function

Source: Japan Mathematical Olympiad Finals, Problem 3

February 7, 2010
functionnumber theory proposednumber theory

Problem Statement

Denote by S(n) S(n) the sum of decimal digits of a positive integer n n. Show that there exist 2002 2002 distinct positive integers n1,n2,⋯ ,n2002 n_1, n_2, \cdots, n_{2002} such that n_1 \plus{} S(n_1) \equal{} n_2 \plus{} S(n_2) \equal{} \cdots \equal{} n_{2002} \plus{} S(n_{2002}).
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