{"status": "success", "data": {"description_md": "Rational numbers $a$ and $b$ are chosen at random among all rational numbers in the interval $[0,2)$ that can be written as fractions $\\frac{n}{d}$ where $n$ and $d$ are integers with $1 \\le d \\le 5$. What is the probability that\n\n$$(\\text{cos}(a\\pi)+i\\text{sin}(b\\pi))^4$$<br>is a real number?\n\n$\\textbf{(A)}\\ \\frac{3}{50} \\qquad\\textbf{(B)}\\ \\frac{4}{25} \\qquad\\textbf{(C)}\\ \\frac{41}{200} \\qquad\\textbf{(D)}\\ \\frac{6}{25} \\qquad\\textbf{(E)}\\ \\frac{13}{50}$\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>Rational numbers  <span class=\"katex--inline\">a</span>  and  <span class=\"katex--inline\">b</span>  are chosen at random among all rational numbers in the interval  <span class=\"katex--inline\">[0,2)</span>  that can be written as fractions  <span class=\"katex--inline\">\\frac{n}{d}</span>  where  <span class=\"katex--inline\">n</span>  and  <span class=\"katex--inline\">d</span>  are integers with  <span class=\"katex--inline\">1 \\le d \\le 5</span> . What is the probability that</p>&#10;<p> <span class=\"katex--display\">(\\text{cos}(a\\pi)+i\\text{sin}(b\\pi))^4</span> <br/>is a real number?</p>&#10;<p> <span class=\"katex--inline\">\\textbf{(A)}\\ \\frac{3}{50} \\qquad\\textbf{(B)}\\ \\frac{4}{25} \\qquad\\textbf{(C)}\\ \\frac{41}{200} \\qquad\\textbf{(D)}\\ \\frac{6}{25} \\qquad\\textbf{(E)}\\ \\frac{13}{50}</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": "2015 AMC 12A Problem 24", "can_next": true, "can_prev": true, "nxt": "/problem/15_amc12A_p25", "prev": "/problem/15_amc12A_p23"}}