{"status": "success", "data": {"description_md": "Triangle $ABC$ is an isosceles right triangle with $AB=AC=3$. Let $M$ be the midpoint of hypotenuse $\\overline{BC}$. Points $I$ and $E$ lie on sides $\\overline{AC}$ and $\\overline{AB}$, respectively, so that $AI>AE$ and $AIME$ is a cyclic quadrilateral. Given that triangle $EMI$ has area $2$, the length $CI$ can be written as $\\frac{a-\\sqrt{b}}{c}$, where $a$, $b$, and $c$ are positive integers and $b$ is not divisible by the square of any prime. What is the value of $a+b+c$?\n\n$\\textbf{(A) }9 \\qquad<br>\\textbf{(B) }10 \\qquad<br>\\textbf{(C) }11 \\qquad<br>\\textbf{(D) }12 \\qquad<br>\\textbf{(E) }13 \\qquad$\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>Triangle  <span class=\"katex--inline\">ABC</span>  is an isosceles right triangle with  <span class=\"katex--inline\">AB=AC=3</span> . Let  <span class=\"katex--inline\">M</span>  be the midpoint of hypotenuse  <span class=\"katex--inline\">\\overline{BC}</span> . Points  <span class=\"katex--inline\">I</span>  and  <span class=\"katex--inline\">E</span>  lie on sides  <span class=\"katex--inline\">\\overline{AC}</span>  and  <span class=\"katex--inline\">\\overline{AB}</span> , respectively, so that  <span class=\"katex--inline\">AI&gt;AE</span>  and  <span class=\"katex--inline\">AIME</span>  is a cyclic quadrilateral. Given that triangle  <span class=\"katex--inline\">EMI</span>  has area  <span class=\"katex--inline\">2</span> , the length  <span class=\"katex--inline\">CI</span>  can be written as  <span class=\"katex--inline\">\\frac{a-\\sqrt{b}}{c}</span> , where  <span class=\"katex--inline\">a</span> ,  <span class=\"katex--inline\">b</span> , and  <span class=\"katex--inline\">c</span>  are positive integers and  <span class=\"katex--inline\">b</span>  is not divisible by the square of any prime. What is the value of  <span class=\"katex--inline\">a+b+c</span> ?</p>&#10;<p> <span class=\"katex--inline\">\\textbf{(A) }9 \\qquad\\textbf{(B) }10 \\qquad\\textbf{(C) }11 \\qquad\\textbf{(D) }12 \\qquad\\textbf{(E) }13 \\qquad</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": "2018 AMC 12A Problem 20", "can_next": true, "can_prev": true, "nxt": "/problem/18_amc12A_p21", "prev": "/problem/18_amc12A_p19"}}