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Expression with sum of digits (bounded?)

Source: Argentina IMO TST 2007 problem 6

August 27, 2009
limitalgebrapolynomialnumber theory unsolvednumber theory

Problem Statement

For natural n n we define s(n) s(n) as the sum of digits of n n (in base ten) Does there exist a positive real constant c c such that for all natural n n we have s(n)s(n2)≤c \frac{s(n)}{s(n^2)} \le c ?
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