18

Part of 2003 Romania Team Selection Test

Problems(1)

For n the sum of digits =d(n), then x+d(x)=m has k solutions

Source: Romanian IMO Team Selection Test TST 2003, problem 18

For every positive integer

n

d(n)

the sum of its digits in the decimal representation. Prove that for each positive integer

k

there exists a positive integer

m

such that the equation

x+d(x)=m

k

solutions in the set of positive integers.

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