{"status": "success", "data": {"description_md": "Let $F(z)=\\frac{z+i}{z-i}$ for all complex numbers $z\\neq i,$ and let $z_n=F(z_{n-1})$ for all positive integers $n.$ Given that $z_0=\\frac 1{137}+i$ and $z_{2002}=a+bi,$ where $a$ and $b$ are real numbers, find $a+b.$\n___\nLeading zeroes must be inputted, so if your answer is `34`, then input `034`. Full 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\">F(z)=\\frac{z+i}{z-i}</span> for all complex numbers <span class=\"katex--inline\">z\\neq i,</span> and let <span class=\"katex--inline\">z_n=F(z_{n-1})</span> for all positive integers <span class=\"katex--inline\">n.</span> Given that <span class=\"katex--inline\">z_0=\\frac 1{137}+i</span> and <span class=\"katex--inline\">z_{2002}=a+bi,</span> where <span class=\"katex--inline\">a</span> and <span class=\"katex--inline\">b</span> are real numbers, find <span class=\"katex--inline\">a+b.</span></p>&#10;<hr/>&#10;<p>Leading zeroes must be inputted, so if your answer is <code>34</code>, then input <code>034</code>. 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": "2002 AIME I Problem 12", "can_next": true, "can_prev": true, "nxt": "/problem/02_aime_I_p13", "prev": "/problem/02_aime_I_p11"}}