{"status": "success", "data": {"description_md": "Each of 2010 boxes in a line contains a single red marble, and for $1 \\le k \\le 2010$, the box in the $k\\text{th}$ position also contains $k$ white marbles. Isabella begins at the first box and successively draws a single marble at random from each box, in order. She stops when she first draws a red marble. Let $P(n)$ be the probability that Isabella stops after drawing exactly $n$ marbles. What is the smallest value of $n$ for which $P(n) < \\frac{1}{2010}$?\n\n$\\textbf{(A)}\\ 45 \\qquad \\textbf{(B)}\\ 63 \\qquad \\textbf{(C)}\\ 64 \\qquad \\textbf{(D)}\\ 201 \\qquad \\textbf{(E)}\\ 1005$\n___\nFull credit goes to [MAA](https://maa.org/) for authoring these problems. These problems were taken on the [AOPS](https://artofproblemsolving.com/) website.", "description_html": "<p>Each of 2010 boxes in a line contains a single red marble, and for  <span class=\"katex--inline\">1 \\le k \\le 2010</span> , the box in the  <span class=\"katex--inline\">k\\text{th}</span>  position also contains  <span class=\"katex--inline\">k</span>  white marbles. Isabella begins at the first box and successively draws a single marble at random from each box, in order. She stops when she first draws a red marble. Let  <span class=\"katex--inline\">P(n)</span>  be the probability that Isabella stops after drawing exactly  <span class=\"katex--inline\">n</span>  marbles. What is the smallest value of  <span class=\"katex--inline\">n</span>  for which  <span class=\"katex--inline\">P(n) &lt; \\frac{1}{2010}</span> ?</p>&#10;<p> <span class=\"katex--inline\">\\textbf{(A)}\\ 45 \\qquad \\textbf{(B)}\\ 63 \\qquad \\textbf{(C)}\\ 64 \\qquad \\textbf{(D)}\\ 201 \\qquad \\textbf{(E)}\\ 1005</span> </p>&#10;<hr><p>Full credit goes to <a href=\"https://maa.org/\">MAA</a> for authoring these problems. These problems were taken on the <a href=\"https://artofproblemsolving.com/\">AOPS</a> website.</p>", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 3, "problem_name": "2010 AMC 12A Problem 19", "can_next": true, "can_prev": true, "nxt": "/problem/10_amc12A_p20", "prev": "/problem/10_amc12A_p18"}}