{"status": "success", "data": {"description_md": "Let $x_0,x_1,x_2,\\ldots$ be a sequence of numbers, where each $x_k$ is either $0$ or $1$. For each positive integer $n$, define \n\n$$S_n = \\sum_{k=0}^{n-1} x_k 2^k$$<br>Suppose $7S_n \\equiv 1 \\pmod{2^n}$ for all $n \\geqslant 1$. What is the value of the sum\n\n$$x_{2019} + 2x_{2020} + 4x_{2021} + 8x_{2022}?$$\n\n$\\textbf{(A) } 6 \\qquad \\textbf{(B) } 7 \\qquad \\textbf{(C) }12\\qquad \\textbf{(D) } 14\\qquad \\textbf{(E) }15$\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>Let <span class=\"katex--inline\">x_0,x_1,x_2,\\ldots</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</p>&#10;<p><span class=\"katex--display\">S_n = \\sum_{k=0}^{n-1} x_k 2^k</span><br/>Suppose <span class=\"katex--inline\">7S_n \\equiv 1 \\pmod{2^n}</span> for all <span class=\"katex--inline\">n \\geqslant 1</span>. What is the value of the sum</p>&#10;<p><span class=\"katex--display\">x_{2019} + 2x_{2020} + 4x_{2021} + 8x_{2022}?</span></p>&#10;<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>&#10;<hr/>&#10;<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>&#10;", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 5, "problem_name": "2022 AMC 12B Problem 23", "can_next": true, "can_prev": true, "nxt": "/problem/22_amc12B_p24", "prev": "/problem/22_amc12B_p22"}}