{"status": "success", "data": {"description_md": "Let $x_0,x_1,x_2,\\dotsc$ be a sequence of numbers, where each $x_k$ is either $0$ or $1$. For each positive integer $n$, define \n$$S_n = \\sum_{k=0}^{n-1} x_k 2^k$$\nSuppose $7S_n \\equiv 1 \\pmod{2^n}$ for all $n \\geq 1$. What is the value of the sum  \n$$x_{2019} + 2x_{2020} + 4x_{2021} + 8x_{2022}?$$\n\n$\\textbf{(A) } 6 \\qquad \\textbf{(B) } 7 \\qquad \\textbf{(C) }12\\qquad \\textbf{(D) } 14\\qquad \\textbf{(E) }15$", "description_html": "<p>Let  <span class=\"katex--inline\">x_0,x_1,x_2,\\dotsc</span>  be a sequence of numbers, where each  <span class=\"katex--inline\">x_k</span>  is either  <span class=\"katex--inline\">0</span>  or  <span class=\"katex--inline\">1</span> . For each positive integer  <span class=\"katex--inline\">n</span> , define<br/>\n <span class=\"katex--display\">S_n = \\sum_{k=0}^{n-1} x_k 2^k</span> <br/>\nSuppose  <span class=\"katex--inline\">7S_n \\equiv 1 \\pmod{2^n}</span>  for all  <span class=\"katex--inline\">n \\geq 1</span> . What is the value of the sum<br/>\n <span class=\"katex--display\">x_{2019} + 2x_{2020} + 4x_{2021} + 8x_{2022}?</span> </p>\n<p> <span class=\"katex--inline\">\\textbf{(A) } 6 \\qquad \\textbf{(B) } 7 \\qquad \\textbf{(C) }12\\qquad \\textbf{(D) } 14\\qquad \\textbf{(E) }15</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": "2022 AMC 10B Problem 25", "can_next": false, "can_prev": true, "nxt": "", "prev": "/problem/22_amc10B_p24"}}