{"status": "success", "data": {"description_md": "Real numbers $a$ and $b$ are chosen with $1<a<b$ such that no triangles with positive area has side lengths $1, a,$ and $b$ or $\\tfrac{1}{b}, \\tfrac{1}{a},$ and $1$. What is the smallest possible value of $b$?\n\n$\\textbf{(A)}\\ \\frac{3+\\sqrt{3}}{2}\\qquad\\textbf{(B)}\\ \\frac{5}{2}\\qquad\\textbf{(C)}\\ \\frac{3+\\sqrt{5}}{2}\\qquad\\textbf{(D)}\\ \\frac{3+\\sqrt{6}}{2}\\qquad\\textbf{(E)}\\ 3$\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>Real numbers  <span class=\"katex--inline\">a</span>  and  <span class=\"katex--inline\">b</span>  are chosen with  <span class=\"katex--inline\">1&lt;a&lt;b</span>  such that no triangles with positive area has side lengths  <span class=\"katex--inline\">1, a,</span>  and  <span class=\"katex--inline\">b</span>  or  <span class=\"katex--inline\">\\tfrac{1}{b}, \\tfrac{1}{a},</span>  and  <span class=\"katex--inline\">1</span> . What is the smallest possible value of  <span class=\"katex--inline\">b</span> ?</p>&#10;<p> <span class=\"katex--inline\">\\textbf{(A)}\\ \\frac{3+\\sqrt{3}}{2}\\qquad\\textbf{(B)}\\ \\frac{5}{2}\\qquad\\textbf{(C)}\\ \\frac{3+\\sqrt{5}}{2}\\qquad\\textbf{(D)}\\ \\frac{3+\\sqrt{6}}{2}\\qquad\\textbf{(E)}\\ 3</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": "2014 AMC 12B Problem 13", "can_next": true, "can_prev": true, "nxt": "/problem/14_amc12B_p14", "prev": "/problem/14_amc12B_p12"}}