{"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$\\textbf{(A)}\\ 45 \\qquad \\textbf{(B)}\\ 63 \\qquad \\textbf{(C)}\\ 64 \\qquad \\textbf{(D)}\\ 201 \\qquad \\textbf{(E)}\\ 1005$", "description_html": "<p>Each of <span class=\"katex--inline\">2010</span> 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;", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 4, "problem_name": "2010 AMC 10A Problem 23", "can_next": true, "can_prev": true, "nxt": "/problem/10_amc10A_p24", "prev": "/problem/10_amc10A_p22"}}