{"status": "success", "data": {"description_md": "A region $S$ in the complex plane is defined by\n\n$$S = \\{x + iy: - 1\\le x\\le1, - 1\\le y\\le1\\}.$$<br>A complex number $z = x + iy$ is chosen uniformly at random from $S$. What is the probability that $\\left(\\frac34 + \\frac34i\\right)z$ is also in $S$?\n\n$\\textbf{(A)}\\ \\frac12\\qquad \\textbf{(B)}\\ \\frac23\\qquad \\textbf{(C)}\\ \\frac34\\qquad \\textbf{(D)}\\ \\frac79\\qquad \\textbf{(E)}\\ \\frac78$\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 region  <span class=\"katex--inline\">S</span>  in the complex plane is defined by</p>&#10;<p> <span class=\"katex--display\">S = \\{x + iy: - 1\\le x\\le1, - 1\\le y\\le1\\}.</span> <br/>A complex number  <span class=\"katex--inline\">z = x + iy</span>  is chosen uniformly at random from  <span class=\"katex--inline\">S</span> . What is the probability that  <span class=\"katex--inline\">\\left(\\frac34 + \\frac34i\\right)z</span>  is also in  <span class=\"katex--inline\">S</span> ?</p>&#10;<p> <span class=\"katex--inline\">\\textbf{(A)}\\ \\frac12\\qquad \\textbf{(B)}\\ \\frac23\\qquad \\textbf{(C)}\\ \\frac34\\qquad \\textbf{(D)}\\ \\frac79\\qquad \\textbf{(E)}\\ \\frac78</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": "2009 AMC 12B Problem 23", "can_next": true, "can_prev": true, "nxt": "/problem/09_amc12B_p24", "prev": "/problem/09_amc12B_p22"}}