{"status": "success", "data": {"description_md": "Let $S$ be the set of lattice points in the coordinate plane, both of whose coordinates are integers between $1$ and $30$, inclusive. Exactly $300$ points in $S$ lie on or below a line with equation $y = mx$. The possible values of $m$ lie in an interval of length $\\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. What is $a + b ?$\n\n$\\textbf{(A)} ~31 \\qquad\\textbf{(B)} ~47 \\qquad\\textbf{(C)} ~62 \\qquad\\textbf{(D)} ~72 \\qquad\\textbf{(E)} ~85$", "description_html": "<p>Let  <span class=\"katex--inline\">S</span>  be the set of lattice points in the coordinate plane, both of whose coordinates are integers between  <span class=\"katex--inline\">1</span>  and  <span class=\"katex--inline\">30</span> , inclusive. Exactly  <span class=\"katex--inline\">300</span>  points in  <span class=\"katex--inline\">S</span>  lie on or below a line with equation  <span class=\"katex--inline\">y = mx</span> . The possible values of  <span class=\"katex--inline\">m</span>  lie in an interval of length  <span class=\"katex--inline\">\\frac{a}{b}</span> , where  <span class=\"katex--inline\">a</span>  and  <span class=\"katex--inline\">b</span>  are relatively prime positive integers. What is  <span class=\"katex--inline\">a + b ?</span> </p>\n<p> <span class=\"katex--inline\">\\textbf{(A)} ~31 \\qquad\\textbf{(B)} ~47 \\qquad\\textbf{(C)} ~62 \\qquad\\textbf{(D)} ~72 \\qquad\\textbf{(E)} ~85</span> </p>\n<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": 4, "problem_name": "2021 AMC 10B Problem 25", "can_next": false, "can_prev": true, "nxt": "", "prev": "/problem/21_amc10B_p24"}}