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Selection and Training Session

Belarus number theory

Problem

Find all pairs

(m, n)

of nonnegative integers for which

m^{2} + 2 \cdot 3^{n} = m (2^{n + 1} - 1) .

Solution

2. See IMO-2010 Shortlist, Problem N2.

Final answer

((m,n) = (6,3), (9,3), (9,5), (54,5))

Techniques

Techniques: modulo, size analysis, order analysis, inequalities