{"status": "success", "data": {"description_md": "How many three-digit positive integers $\\underline{a} \\ \\underline{b} \\ \\underline{c}$ are there whose nonzero digits $a,b,$ and $c$ satisfy\n$$0.\\overline{\\underline{a}~\\underline{b}~\\underline{c}} = \\frac{1}{3} (0.\\overline{a} + 0.\\overline{b} + 0.\\overline{c})?$$\n\n(The bar indicates repetition, thus $0.\\overline{\\underline{a}~\\underline{b}~\\underline{c}}$ is the infinite repeating decimal $0.\\underline{a}~\\underline{b}~\\underline{c}~\\underline{a}~\\underline{b}~\\underline{c}~\\cdots$)\n\n$\\textbf{(A) } 9 \\qquad \\textbf{(B) } 10 \\qquad \\textbf{(C) } 11 \\qquad \\textbf{(D) } 13 \\qquad \\textbf{(E) } 14$", "description_html": "<p>How many three-digit positive integers <span class=\"katex--inline\">\\underline{a} \\ \\underline{b} \\ \\underline{c}</span> are there whose nonzero digits <span class=\"katex--inline\">a,b,</span> and <span class=\"katex--inline\">c</span> satisfy<br/>&#10;<span class=\"katex--display\">0.\\overline{\\underline{a}~\\underline{b}~\\underline{c}} = \\frac{1}{3} (0.\\overline{a} + 0.\\overline{b} + 0.\\overline{c})?</span></p>&#10;<p>(The bar indicates repetition, thus <span class=\"katex--inline\">0.\\overline{\\underline{a}~\\underline{b}~\\underline{c}}</span> is the infinite repeating decimal <span class=\"katex--inline\">0.\\underline{a}~\\underline{b}~\\underline{c}~\\underline{a}~\\underline{b}~\\underline{c}~\\cdots</span>)</p>&#10;<p><span class=\"katex--inline\">\\textbf{(A) } 9 \\qquad \\textbf{(B) } 10 \\qquad \\textbf{(C) } 11 \\qquad \\textbf{(D) } 13 \\qquad \\textbf{(E) } 14</span></p>&#10;", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 3, "problem_name": "2022 AMC 10A Problem 17", "can_next": true, "can_prev": true, "nxt": "/problem/22_amc10A_p18", "prev": "/problem/22_amc10A_p16"}}