26th Junior Balkan Mathematical Olympiad | JBMO 2022 - Problem 1

Problem 1

Find all pairs of positive integers $(a, b)$ such that $11ab \le a^3 - b^3 \le 12ab.$

Solution

Clearly, $a>b$ .

If $a\ge b+4$ , then $a^3-b^3-12ab=3ab(a-b-4)+(a-b)^3>0$ , contradiction.

If $a\leq b+2$ , then $a^3-b^3-11ab=(a-b)^3-ab(11-3(a-b))\leq 8-5ab\leq -2<0$ , contradiction.

Hence, $a=b+3$ . This gives us $11b(b+3)\leq (b+3)^3-b^3\leq 12b(b+3)\Leftrightarrow 9\leq b(b+3)\leq \frac{27}2\Rightarrow b=2$
So the only solution is $(a,b)=(5,2)$ .