{"status": "success", "data": {"description_md": "For how many values of $a$ is it true that the line $y = x + a$ passes through the<br>vertex of the parabola $y = x^2 + a^2$ ?\n\n$\\mathrm{(A)}\\ 0      \\qquad<br>\\mathrm{(B)}\\ 1      \\qquad<br>\\mathrm{(C)}\\ 2      \\qquad<br>\\mathrm{(D)}\\ 10      \\qquad<br>\\mathrm{(E)}\\ \\text{infinitely many}$\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>For how many values of  <span class=\"katex--inline\">a</span>  is it true that the line  <span class=\"katex--inline\">y = x + a</span>  passes through the<br/>vertex of the parabola  <span class=\"katex--inline\">y = x^2 + a^2</span>  ?</p>&#10;<p> <span class=\"katex--inline\">\\mathrm{(A)}\\ 0      \\qquad\\mathrm{(B)}\\ 1      \\qquad\\mathrm{(C)}\\ 2      \\qquad\\mathrm{(D)}\\ 10      \\qquad\\mathrm{(E)}\\ \\text{infinitely many}</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": 2, "problem_name": "2005 AMC 12B Problem 8", "can_next": true, "can_prev": true, "nxt": "/problem/05_amc12B_p09", "prev": "/problem/05_amc12B_p07"}}