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SELECTION and TRAINING SESSION

Belarus number theory

Problem

Determine all pairs

(x, y)

of positive integers such that

3 7 x^{2} - 13 x y + 7 y^{2} = ∣ x - y ∣ + 1.

Solution

2. See IMO-2014 Shortlist, Problem N2.

Final answer

{(x, y) = (1, 1)} ∪ { (x, y) = (n^3 + 2n^2 − n − 1, n^3 + n^2 − 2n − 1), (n^3 + n^2 − 2n − 1, n^3 + 2n^2 − n − 1) for integers n ≥ 2 }

Techniques

Techniques: modulo, size analysis, order analysis, inequalities