{"status": "success", "data": {"description_md": "A sequence $(a_1,b_1)$, $(a_2,b_2)$, $(a_3,b_3)$, $\\ldots$ of points in the coordinate plane satisfies\n\n$(a_{n + 1}, b_{n + 1}) = (\\sqrt {3}a_n - b_n, \\sqrt {3}b_n + a_n)$  for $n = 1,2,3,\\ldots$.<br>Suppose that $(a_{100},b_{100}) = (2,4)$.  What is $a_1 + b_1$?\n\n$\\mathrm{(A)}\\ -\\frac{1}{2^{97}}\\qquad\\mathrm{(B)}\\ -\\frac{1}{2^{99}}\\qquad\\mathrm{(C)}\\ 0\\qquad\\mathrm{(D)}\\ \\frac{1}{2^{98}}\\qquad\\mathrm{(E)}\\ \\frac{1}{2^{96}}$\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>A sequence  <span class=\"katex--inline\">(a_1,b_1)</span> ,  <span class=\"katex--inline\">(a_2,b_2)</span> ,  <span class=\"katex--inline\">(a_3,b_3)</span> ,  <span class=\"katex--inline\">\\ldots</span>  of points in the coordinate plane satisfies</p>&#10;<p> <span class=\"katex--inline\">(a_{n + 1}, b_{n + 1}) = (\\sqrt {3}a_n - b_n, \\sqrt {3}b_n + a_n)</span>   for  <span class=\"katex--inline\">n = 1,2,3,\\ldots</span> .<br/>Suppose that  <span class=\"katex--inline\">(a_{100},b_{100}) = (2,4)</span> .  What is  <span class=\"katex--inline\">a_1 + b_1</span> ?</p>&#10;<p> <span class=\"katex--inline\">\\mathrm{(A)}\\ -\\frac{1}{2^{97}}\\qquad\\mathrm{(B)}\\ -\\frac{1}{2^{99}}\\qquad\\mathrm{(C)}\\ 0\\qquad\\mathrm{(D)}\\ \\frac{1}{2^{98}}\\qquad\\mathrm{(E)}\\ \\frac{1}{2^{96}}</span> </p>&#10;<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": 5, "problem_name": "2008 AMC 12A Problem 25", "can_next": false, "can_prev": true, "nxt": "", "prev": "/problem/08_amc12A_p24"}}