124 Splits Into Squares — Twelve Pairs Solve This Equation | Math Olympiad

Completing the square, perfect square testing, olympiad number theory — multiply by 4, complete the square, and a bounded search reveals twelve integer pairs hiding behind one symmetric equation. a²+ab+b² = 31 with integer constraints. There's no obvious factorization, so multiply through by 4 and complete the square on a: 4a²+4ab+4b² = 124 rewrites as (2a+b)²+3b² = 124. That single move converts the cross term into a clean sum of two squares structure. Since 3b² can't exceed 124, b is bounded: |b| ≤ 6. That's only 13 values to check. For each one, compute 124−3b² and test whether it's a perfect square. Most values fail immediately — b = 0, ±2, ±3, ±4 all give non-squares. Three values survive: b = ±1 gives 121 = 11², b = ±5 gives 49 = 7², b = ±6 gives 16 = 4². Each surviving b produces two values of a from 2a+b = ±√(124−3b²). Working through all six valid b values gives twelve integer pairs total: (5,1), (1,5), (−6,1), (1,−6), (6,−1), (−1,6), (−5,−1), (−1,−5), (−6,5), (5,−6), (−5,6), (6,−5). The symmetry isn't coincidental. Swapping a and b leaves the equation unchanged, and flipping both signs leaves it unchanged too — that's why the solutions cluster in mirrored groups. Students who try testing integer values of a one by one without the bounding argument might find a couple of pairs but have no systematic way to confirm completeness. Bounding b through the completed square guarantees every solution gets found — and proves nothing is missed. This bounded-search technique — complete the square, derive a bound on one variable, test each candidate for a perfect square — works for any symmetric Diophantine equation of the form a²+ab+b² = k. The bound always comes from the completed square form, and exhausting it guarantees completeness. 🔔 Daily olympiad problems — algebra, number theory, geometry, combinatorics. #OlympiadMath #CompletingTheSquare #DiophantineEquation #NumberTheory #CompetitionMath #AMCPrep #AIМЕPrep #MathOlympiad #IntegerSolutions #BoundedSearch #PerfectSquareTest #OlympiadNumberTheory