{"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, $\\overline{AB}$ 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$\\textbf{(A) }\\frac{7}{22}\\qquad\\textbf{(B) }\\frac{1}{3}\\qquad\\textbf{(C) }\\frac{3}{8}\\qquad\\textbf{(D) }\\frac{5}{12}\\qquad\\textbf{(E) }\\frac{1}{2}$", "description_html": "

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, \\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) ?

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\\textbf{(A) }\\frac{7}{22}\\qquad\\textbf{(B) }\\frac{1}{3}\\qquad\\textbf{(C) }\\frac{3}{8}\\qquad\\textbf{(D) }\\frac{5}{12}\\qquad\\textbf{(E) }\\frac{1}{2}

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Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.

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