{"status": "success", "data": {"description_md": "Let $\\overline{MN}$ be a diameter of a circle with diameter $1$. Let $A$ and $B$ be points on one of the semicircular arcs determined by $\\overline{MN}$ such that $A$ is the midpoint of the semicircle and $MB=\\frac35$. Point $C$ lies on the other semicircular arc. Let $d$ be the length of the line segment whose endpoints are the intersections of diameter $\\overline{MN}$ with the chords $\\overline{AC}$ and $\\overline{BC}$. The largest possible value of $d$ can be written in the form $r-s\\sqrt{t}$, where $r$, $s$, and $t$ are positive integers and $t$ is not divisible by the square of any prime. Find $r+s+t$.\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": "<p>Let <span class=\"katex--inline\">\\overline{MN}</span> be a diameter of a circle with diameter <span class=\"katex--inline\">1</span>. Let <span class=\"katex--inline\">A</span> and <span class=\"katex--inline\">B</span> be points on one of the semicircular arcs determined by <span class=\"katex--inline\">\\overline{MN}</span> such that <span class=\"katex--inline\">A</span> is the midpoint of the semicircle and <span class=\"katex--inline\">MB=\\frac35</span>. Point <span class=\"katex--inline\">C</span> lies on the other semicircular arc. Let <span class=\"katex--inline\">d</span> be the length of the line segment whose endpoints are the intersections of diameter <span class=\"katex--inline\">\\overline{MN}</span> with the chords <span class=\"katex--inline\">\\overline{AC}</span> and <span class=\"katex--inline\">\\overline{BC}</span>. The largest possible value of <span class=\"katex--inline\">d</span> can be written in the form <span class=\"katex--inline\">r-s\\sqrt{t}</span>, where <span class=\"katex--inline\">r</span>, <span class=\"katex--inline\">s</span>, and <span class=\"katex--inline\">t</span> are positive integers and <span class=\"katex--inline\">t</span> is not divisible by the square of any prime. Find <span class=\"katex--inline\">r+s+t</span>.</p>&#10;<hr/>&#10;<p>Leading zeroes must be inputted, so if your answer is <code>34</code>, then input <code>034</code>. 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>&#10;", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 6, "problem_name": "2009 AIME II Problem 15", "can_next": false, "can_prev": true, "nxt": "", "prev": "/problem/09_aime_II_p14"}}