{"status": "success", "data": {"description_md": "Let $R$, $S$, and $T$ be squares that have vertices at lattice points (i.e., points whose coordinates are both integers) in the coordinate plane, together with their interiors. The bottom edge of each square is on the $x$-axis. The left edge of $R$ and the right edge of $S$ are on the $y$-axis, and $R$ contains $\\frac{9}{4}$ as many lattice points as does $S$. The top two vertices of $T$ are in $R \\cup S$, and $T$ contains $\\frac{1}{4}$ of the lattice points contained in $R \\cup S.$ See the figure (not drawn to scale).\n\n<center>\n<img class=\"problem-image\" height=\"288\" src=\"https://latex.artofproblemsolving.com/d/1/9/d19ef900b6330bb16c3488ac1593a01dce4bedb8.png\" width=\"378\"/>\n</center><br>\n\nThe fraction of lattice points in $S$ that are in $S \\cap T$ is $27$ times the fraction of lattice points in $R$ that are in $R \\cap T$. What is the minimum possible value of the edge length of $R$ plus the edge length of $S$ plus the edge length of $T$?\n\n$\\textbf{(A) }336\\qquad\\textbf{(B) }337\\qquad\\textbf{(C) }338\\qquad\\textbf{(D) }339\\qquad\\textbf{(E) }340$", "description_html": "<p>Let <span class=\"katex--inline\">R</span>, <span class=\"katex--inline\">S</span>, and <span class=\"katex--inline\">T</span> be squares that have vertices at lattice points (i.e., points whose coordinates are both integers) in the coordinate plane, together with their interiors. The bottom edge of each square is on the <span class=\"katex--inline\">x</span>-axis. The left edge of <span class=\"katex--inline\">R</span> and the right edge of <span class=\"katex--inline\">S</span> are on the <span class=\"katex--inline\">y</span>-axis, and <span class=\"katex--inline\">R</span> contains <span class=\"katex--inline\">\\frac{9}{4}</span> as many lattice points as does <span class=\"katex--inline\">S</span>. The top two vertices of <span class=\"katex--inline\">T</span> are in <span class=\"katex--inline\">R \\cup S</span>, and <span class=\"katex--inline\">T</span> contains <span class=\"katex--inline\">\\frac{1}{4}</span> of the lattice points contained in <span class=\"katex--inline\">R \\cup S.</span> See the figure (not drawn to scale).</p>&#10;<center>&#10;<img class=\"problem-image\" height=\"288\" src=\"https://latex.artofproblemsolving.com/d/1/9/d19ef900b6330bb16c3488ac1593a01dce4bedb8.png\" width=\"378\"/>&#10;</center><br/>&#10;<p>The fraction of lattice points in <span class=\"katex--inline\">S</span> that are in <span class=\"katex--inline\">S \\cap T</span> is <span class=\"katex--inline\">27</span> times the fraction of lattice points in <span class=\"katex--inline\">R</span> that are in <span class=\"katex--inline\">R \\cap T</span>. What is the minimum possible value of the edge length of <span class=\"katex--inline\">R</span> plus the edge length of <span class=\"katex--inline\">S</span> plus the edge length of <span class=\"katex--inline\">T</span>?</p>&#10;<p><span class=\"katex--inline\">\\textbf{(A) }336\\qquad\\textbf{(B) }337\\qquad\\textbf{(C) }338\\qquad\\textbf{(D) }339\\qquad\\textbf{(E) }340</span></p>&#10;", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 4, "problem_name": "2022 AMC 10A Problem 25", "can_next": false, "can_prev": true, "nxt": "", "prev": "/problem/22_amc10A_p24"}}