{"status": "success", "data": {"description_md": "Let \n\n$S_1=\\{(x,y)|\\log_{10}(1+x^2+y^2)\\le 1+\\log_{10}(x+y)\\}$<br>and \n\n$S_2=\\{(x,y)|\\log_{10}(2+x^2+y^2)\\le 2+\\log_{10}(x+y)\\}$.<br>What is the ratio of the area of $S_2$ to the area of $S_1$?\n\n$\\mathrm{(A) \\ } 98\\qquad \\mathrm{(B) \\ } 99\\qquad \\mathrm{(C) \\ } 100\\qquad \\mathrm{(D) \\ } 101\\qquad \\mathrm{(E) \\ }  102$\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</p>&#10;<p> <span class=\"katex--inline\">S_1=\\{(x,y)|\\log_{10}(1+x^2+y^2)\\le 1+\\log_{10}(x+y)\\}</span> <br/>and</p>&#10;<p> <span class=\"katex--inline\">S_2=\\{(x,y)|\\log_{10}(2+x^2+y^2)\\le 2+\\log_{10}(x+y)\\}</span> .<br/>What is the ratio of the area of  <span class=\"katex--inline\">S_2</span>  to the area of  <span class=\"katex--inline\">S_1</span> ?</p>&#10;<p> <span class=\"katex--inline\">\\mathrm{(A) \\ } 98\\qquad \\mathrm{(B) \\ } 99\\qquad \\mathrm{(C) \\ } 100\\qquad \\mathrm{(D) \\ } 101\\qquad \\mathrm{(E) \\ }  102</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": "2006 AMC 12A Problem 21", "can_next": true, "can_prev": true, "nxt": "/problem/06_amc12A_p22", "prev": "/problem/06_amc12A_p20"}}