{"status": "success", "data": {"description_md": "Let $S$ be the set of points whose coordinates $x,$ $y,$ and $z$ are integers that satisfy $0\\le x\\le2,$ $0\\le y\\le3,$ and $0\\le z\\le4.$ Two distinct points are randomly chosen from $S.$ The probability that the midpoint of the segment they determine also belongs to $S$ is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$\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 S be the set of points whose coordinates x, y, and z are integers that satisfy 0\\le x\\le2, 0\\le y\\le3, and 0\\le z\\le4. Two distinct points are randomly chosen from S. The probability that the midpoint of the segment they determine also belongs to S is m/n, where m and n are relatively prime positive integers. Find m+n.

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": "2001 AIME I Problem 10", "can_next": true, "can_prev": true, "nxt": "/problem/01_aime_I_p11", "prev": "/problem/01_aime_I_p09"}}