{"status": "success", "data": {"description_md": "Let $z_1 = 18 + 83i$, $z_2 = 18 + 39i,$ and $z_3 = 78 + 99i,$ where $i = \\sqrt{-1}$. Let $z$ be the unique complex number with the properties that $\\frac{z_3 - z_1}{z_2 - z_1} \\cdot \\frac{z - z_2}{z - z_3}$ is a real number and the imaginary part of $z$ is the greatest possible. Find the real part of $z$.\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\">z_1 = 18 + 83i</span>, <span class=\"katex--inline\">z_2 = 18 + 39i,</span> and <span class=\"katex--inline\">z_3 = 78 + 99i,</span> where <span class=\"katex--inline\">i = \\sqrt{-1}</span>. Let <span class=\"katex--inline\">z</span> be the unique complex number with the properties that <span class=\"katex--inline\">\\frac{z_3 - z_1}{z_2 - z_1} \\cdot \\frac{z - z_2}{z - z_3}</span> is a real number and the imaginary part of <span class=\"katex--inline\">z</span> is the greatest possible. Find the real part of <span class=\"katex--inline\">z</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": "2017 AIME I Problem 10", "can_next": true, "can_prev": true, "nxt": "/problem/17_aime_I_p11", "prev": "/problem/17_aime_I_p09"}}