{"status": "success", "data": {"description_md": "If $A$ and $B$ are vertices of a polyhedron, define the distance $d(A, B)$ to be the minimum number of edges of the polyhedron one must traverse in order to connect $A$ and $B$. For example, if $\\overline{AB}$ is an edge of the polyhedron, then $d(A, B) = 1$, but if $\\overline{AC}$ and $\\overline{CB}$ are edges and $\\overline{AB}$ is not an edge, then $d(A, B) = 2$. Let $Q$, $R$, and $S$ be randomly chosen distinct vertices of a regular icosahedron (regular polyhedron made up of 20 equilateral triangles). What is the probability that $d(Q, R) > d(R, S)$?\n\n$\\textbf{(A)}~\\frac{7}{22}\\qquad\\textbf{(B)}~\\frac13\\qquad\\textbf{(C)}~\\frac38\\qquad\\textbf{(D)}~\\frac5{12}\\qquad\\textbf{(E)}~\\frac12$\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>If  <span class=\"katex--inline\">A</span>  and  <span class=\"katex--inline\">B</span>  are vertices of a polyhedron, define the distance  <span class=\"katex--inline\">d(A, B)</span>  to be the minimum number of edges of the polyhedron one must traverse in order to connect  <span class=\"katex--inline\">A</span>  and  <span class=\"katex--inline\">B</span> . For example, if  <span class=\"katex--inline\">\\overline{AB}</span>  is an edge of the polyhedron, then  <span class=\"katex--inline\">d(A, B) = 1</span> , but if  <span class=\"katex--inline\">\\overline{AC}</span>  and  <span class=\"katex--inline\">\\overline{CB}</span>  are edges and  <span class=\"katex--inline\">\\overline{AB}</span>  is not an edge, then  <span class=\"katex--inline\">d(A, B) = 2</span> . Let  <span class=\"katex--inline\">Q</span> ,  <span class=\"katex--inline\">R</span> , and  <span class=\"katex--inline\">S</span>  be randomly chosen distinct vertices of a regular icosahedron (regular polyhedron made up of 20 equilateral triangles). What is the probability that  <span class=\"katex--inline\">d(Q, R) &gt; d(R, S)</span> ?</p>&#10;<p> <span class=\"katex--inline\">\\textbf{(A)}~\\frac{7}{22}\\qquad\\textbf{(B)}~\\frac13\\qquad\\textbf{(C)}~\\frac38\\qquad\\textbf{(D)}~\\frac5{12}\\qquad\\textbf{(E)}~\\frac12</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": "2023 AMC 12A Problem 21", "can_next": true, "can_prev": true, "nxt": "/problem/23_amc12A_p22", "prev": "/problem/23_amc12A_p20"}}