{"status": "success", "data": {"description_md": "A sequence $a_1,a_2,\\ldots$ of non-negative integers is defined by the rule $a_{n+2}=|a_{n+1}-a_n|$ for $n\\geq 1$. If $a_1=999$, $a_2<999$ and $a_{2006}=1$, how many different values of $a_2$ are possible?\n\n$\\mathrm{(A)}\\ 165<br>\\qquad<br>\\mathrm{(B)}\\ 324<br>\\qquad<br>\\mathrm{(C)}\\ 495<br>\\qquad<br>\\mathrm{(D)}\\ 499<br>\\qquad<br>\\mathrm{(E)}\\ 660$\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,a_2,\\ldots</span>  of non-negative integers is defined by the rule  <span class=\"katex--inline\">a_{n+2}=|a_{n+1}-a_n|</span>  for  <span class=\"katex--inline\">n\\geq 1</span> . If  <span class=\"katex--inline\">a_1=999</span> ,  <span class=\"katex--inline\">a_2&lt;999</span>  and  <span class=\"katex--inline\">a_{2006}=1</span> , how many different values of  <span class=\"katex--inline\">a_2</span>  are possible?</p>&#10;<p> <span class=\"katex--inline\">\\mathrm{(A)}\\ 165\\qquad\\mathrm{(B)}\\ 324\\qquad\\mathrm{(C)}\\ 495\\qquad\\mathrm{(D)}\\ 499\\qquad\\mathrm{(E)}\\ 660</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": "2006 AMC 12B Problem 25", "can_next": false, "can_prev": true, "nxt": "", "prev": "/problem/06_amc12B_p24"}}