IMO Shortlist 2003 problem C6


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2. travnja 2012.
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Let f(k) be the number of all non-negative integers n satisfying the following conditions:

(1) The integer n has exactly k digits in the decimal representation (where the first digit is not necessarily non-zero!), i. e. we have 0 \leq n <10^k.

(2) These k digits of n can be permuted in such a way that the resulting number is divisible by 11.

Show that for any positive integer number m, we have f\left(2m\right) = 10 f\left(2m - 1\right).
Izvor: Međunarodna matematička olimpijada, shortlist 2003