{"status": "success", "data": {"description_md": "Let $M \\ge 3$ be an integer and let $S = \\{3,4,5,\\ldots,m\\}$. Find the smallest value of $m$ such that for every partition of $S$ into two subsets, at least one of the subsets contains integers $a$, $b$, and $ c$ (not necessarily distinct) such that $ab = c$.

Note: a partition of $S$ is a pair of sets $A$, $B$ such that $A \\cap B = \\emptyset$, $A \\cup B = S$.\n___\nLeading zeroes must be inputted, so if your answer is `34`, then input `034`. Full credit goes to [MAA](https://maa.org/) for authoring these problems. These problems were taken on the [AOPS](https://artofproblemsolving.com/) website.", "description_html": "

Let M \\ge 3 be an integer and let S = \\{3,4,5,\\ldots,m\\}. Find the smallest value of m such that for every partition of S into two subsets, at least one of the subsets contains integers a, b, and $ c$ (not necessarily distinct) such that ab = c.

Note: a partition of S is a pair of sets A, B such that A \\cap B = \\emptyset, A \\cup B = S.

Leading zeroes must be inputted, so if your answer is 34, then input 034. Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.

", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 5, "problem_name": "2010 AIME I Problem 12", "can_next": true, "can_prev": true, "nxt": "/problem/10_aime_I_p13", "prev": "/problem/10_aime_I_p11"}}