{"status": "success", "data": {"description_md": "Which of the following describes the set of values of $a$ for which the curves $x^2+y^2=a^2$ and $y=x^2-a$ in the real $xy$-plane intersect at exactly $3$ points?\n\n$\\textbf{(A) }a=\\frac14 \\qquad<br>\\textbf{(B) }\\frac14 < a < \\frac12 \\qquad<br>\\textbf{(C) }a>\\frac14 \\qquad<br>\\textbf{(D) }a=\\frac12 \\qquad<br>\\textbf{(E) }a>\\frac12 \\qquad$\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>Which of the following describes the set of values of  <span class=\"katex--inline\">a</span>  for which the curves  <span class=\"katex--inline\">x^2+y^2=a^2</span>  and  <span class=\"katex--inline\">y=x^2-a</span>  in the real  <span class=\"katex--inline\">xy</span> -plane intersect at exactly  <span class=\"katex--inline\">3</span>  points?</p>&#10;<p> <span class=\"katex--inline\">\\textbf{(A) }a=\\frac14 \\qquad\\textbf{(B) }\\frac14 &lt; a &lt; \\frac12 \\qquad\\textbf{(C) }a&gt;\\frac14 \\qquad\\textbf{(D) }a=\\frac12 \\qquad\\textbf{(E) }a&gt;\\frac12 \\qquad</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": 3, "problem_name": "2018 AMC 12A Problem 16", "can_next": true, "can_prev": true, "nxt": "/problem/18_amc12A_p17", "prev": "/problem/18_amc12A_p15"}}