Number theory · Gcd and lcm · Cyclic sums · Divisibility · Inequalities
Suppose that pairwise different positive integers \(a_1, a_2, \dots, a_{2024}\) satisfy
\[ [a_1, a_2] + (a_2, a_3) + [a_3, a_4] + (a_4, a_5) + \dots + [a_{2023}, a_{2024}] + (a_{2024}, a_1) = a_1 + a_2 + \dots + a_{2024} . \]
Prove the inequality
\[ (a_1, a_2) + [a_2, a_3] + (a_3, a_4) + [a_4, a_5] + \dots + (a_{2023}, a_{2024}) + [a_{2024}, a_1] \geq a_1 + a_2 + \dots + a_{2024} . \]
Is it possible that equality holds in this inequality? (For positive integers \(a\) and \(b\), the numbers \((a, b)\) and \([a, b]\) denote their greatest common divisor and their least common multiple, respectively.)
Sign in to check answers, open hints, read the full solution, and track your progress. Statements are always free.
Serbian National Competition (Drzavno takmicenje) 2024, high school grade I, category A, problem 3. Organized by the Mathematical Society of Serbia (DMS). Source