{"status": "success", "data": {"description_md": "For positive integers $n$, let $\\tau (n)$ denote the number of positive integer divisors of $n$, including $1$ and $n$. For example, $\\tau (1)=1$ and $\\tau(6) =4$. Define $S(n)$ by $$S(n)=\\tau(1)+ \\tau(2) + ... + \\tau(n). $$Let $a$ denote the number of positive integers $n \\leq 2005$ with $S(n)$ odd, and let $b$ denote the number of positive integers $n \\leq 2005$ with $S(n)$ even. Find $|a-b|$.\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": "

For positive integers n, let \\tau (n) denote the number of positive integer divisors of n, including 1 and n. For example, \\tau (1)=1 and \\tau(6) =4. Define S(n) by S(n)=\\tau(1)+ \\tau(2) + ... + \\tau(n).Let a denote the number of positive integers n \\leq 2005 with S(n) odd, and let b denote the number of positive integers n \\leq 2005 with S(n) even. Find |a-b|.

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": "2005 AIME I Problem 12", "can_next": true, "can_prev": true, "nxt": "/problem/05_aime_I_p13", "prev": "/problem/05_aime_I_p11"}}