Number theory · Gcd · Divisibility · Inequalities from divisibility · Consecutive integers
Let \(a\), \(b\), \(c\) be arbitrary positive integers. Prove the inequality
\[ \gcd(a,\,b-1)\cdot\gcd(b,\,c-1)\cdot\gcd(c,\,a-1) \;\le\; ab+bc+ca-a-b-c+1 , \]
and prove that equality is attained for infinitely many triples \((a,b,c)\). (As usual \(\gcd(x,0)=x\); this case occurs when one of \(a\), \(b\), \(c\) equals \(1\).)
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Serbian National Competition (Drzavno takmicenje) 2015, high school grade I, category A, problem 1. Organized by the Mathematical Society of Serbia (DMS). Source