{"status": "success", "data": {"description_md": "Let $w = \\dfrac{\\sqrt{3} + i}{2}$ and $z = \\dfrac{-1 + i\\sqrt{3}}{2},$ where $i = \\sqrt{-1}.$ Find the number of ordered pairs $(r,s)$ of positive integers not exceeding $100$ that satisfy the equation $i \\cdot w^r = z^s.$<br>Leading zeroes must be inputted, so if your answer is `34`, then input `034`\n\n___\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\">w = \\dfrac{\\sqrt{3} + i}{2}</span> and <span class=\"katex--inline\">z = \\dfrac{-1 + i\\sqrt{3}}{2},</span> where <span class=\"katex--inline\">i = \\sqrt{-1}.</span> Find the number of ordered pairs <span class=\"katex--inline\">(r,s)</span> of positive integers not exceeding <span class=\"katex--inline\">100</span> that satisfy the equation <span class=\"katex--inline\">i \\cdot w^r = z^s.</span><br/>Leading zeroes must be inputted, so if your answer is <code>34</code>, then input <code>034</code></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": 3, "problem_name": "2022 AIME I Problem 4", "can_next": true, "can_prev": true, "nxt": "/problem/22_aime_I_p05", "prev": "/problem/22_aime_I_p03"}}