Find all functions f: N N, such that mf(m) + n is divisible by m^2 + f(n) for all m, n N.

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Selection and Training Session

Belarus algebra

Problem

Find all functions

f : N \to N

, such that

m f (m) + n

is divisible by

m^{2} + f (n)

for all

m, n \in N

Solution

### 2. See IMO-2013 Shortlist, Problem N1.

Final answer

f(n) = n for all n in the positive integers

Techniques

Existential quantifiersTechniques: modulo, size analysis, order analysis, inequalities