{"status": "success", "data": {"description_md": "A lattice point in an $xy$-coordinate system is any point $(x, y)$ where both $x$ and $y$ are integers. The graph of $y = mx + 2$ passes through no lattice point with $0 < x \\leq 100$ for all $m$ such that $\\frac{1}{2} < m < a$. What is the maximum possible value of $a$?\n\n$\\textbf{(A)}\\ \\frac{51}{101} \\qquad \\textbf{(B)}\\ \\frac{50}{99} \\qquad \\textbf{(C)}\\ \\frac{51}{100} \\qquad \\textbf{(D)}\\ \\frac{52}{101} \\qquad \\textbf{(E)}\\ \\frac{13}{25}$\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>A lattice point in an  <span class=\"katex--inline\">xy</span> -coordinate system is any point  <span class=\"katex--inline\">(x, y)</span>  where both  <span class=\"katex--inline\">x</span>  and  <span class=\"katex--inline\">y</span>  are integers. The graph of  <span class=\"katex--inline\">y = mx + 2</span>  passes through no lattice point with  <span class=\"katex--inline\">0 &lt; x \\leq 100</span>  for all  <span class=\"katex--inline\">m</span>  such that  <span class=\"katex--inline\">\\frac{1}{2} &lt; m &lt; a</span> . What is the maximum possible value of  <span class=\"katex--inline\">a</span> ?</p>&#10;<p> <span class=\"katex--inline\">\\textbf{(A)}\\ \\frac{51}{101} \\qquad \\textbf{(B)}\\ \\frac{50}{99} \\qquad \\textbf{(C)}\\ \\frac{51}{100} \\qquad \\textbf{(D)}\\ \\frac{52}{101} \\qquad \\textbf{(E)}\\ \\frac{13}{25}</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": "2011 AMC 12B Problem 19", "can_next": true, "can_prev": true, "nxt": "/problem/11_amc12B_p20", "prev": "/problem/11_amc12B_p18"}}