{"status": "success", "data": {"description_md": "Let $S=\\{(x,y) : x\\in \\{0,1,2,3,4\\}, y\\in \\{0,1,2,3,4,5\\},\\text{ and } (x,y)\\ne (0,0)\\}$. <br>Let $T$ be the set of all right triangles whose vertices are in $S$. For every right triangle $t=\\triangle{ABC}$ with vertices $A$, $B$, and $C$ in counter-clockwise order and right angle at $A$, let $f(t)=\\tan(\\angle{CBA})$. What is $$\\prod_{t\\in T} f(t)?$$\n\n$\\textbf{(A)}\\ 1\\qquad\\textbf{(B)}\\ \\frac{625}{144}\\qquad\\textbf{(C)}\\ \\frac{125}{24}\\qquad\\textbf{(D)}\\ 6\\qquad\\textbf{(E)}\\ \\frac{625}{24}$\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>Let  <span class=\"katex--inline\">S=\\{(x,y) : x\\in \\{0,1,2,3,4\\}, y\\in \\{0,1,2,3,4,5\\},\\text{ and } (x,y)\\ne (0,0)\\}</span> . <br/>Let  <span class=\"katex--inline\">T</span>  be the set of all right triangles whose vertices are in  <span class=\"katex--inline\">S</span> . For every right triangle  <span class=\"katex--inline\">t=\\triangle{ABC}</span>  with vertices  <span class=\"katex--inline\">A</span> ,  <span class=\"katex--inline\">B</span> , and  <span class=\"katex--inline\">C</span>  in counter-clockwise order and right angle at  <span class=\"katex--inline\">A</span> , let  <span class=\"katex--inline\">f(t)=\\tan(\\angle{CBA})</span> . What is  <span class=\"katex--display\">\\prod_{t\\in T} f(t)?</span> </p>&#10;<p> <span class=\"katex--inline\">\\textbf{(A)}\\ 1\\qquad\\textbf{(B)}\\ \\frac{625}{144}\\qquad\\textbf{(C)}\\ \\frac{125}{24}\\qquad\\textbf{(D)}\\ 6\\qquad\\textbf{(E)}\\ \\frac{625}{24}</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": 5, "problem_name": "2012 AMC 12B Problem 25", "can_next": false, "can_prev": true, "nxt": "", "prev": "/problem/12_amc12B_p24"}}