{"status": "success", "data": {"description_md": "Let $\\mathcal{P}$ be the image when the parabola $y=x^2$ is rotated $45^\\circ$ clockwise around the origin. Then, the equation $x^2+y^2-\\sqrt{2}x-\\sqrt{2}y=k$ intersects $\\mathcal{P}$ at more than $2$ points. The values of $k$ that satisfy this form a half-open interval with length $\\ell$. What is $\\ell$?\n\n$\\textbf{(A)}~\\tfrac{1}{4}\\qquad\\textbf{(B)}~\\tfrac{1}{2}\\qquad\\textbf{(C)}~1\\qquad\\textbf{(D)}~2\\qquad\\textbf{(E)}~4$", "description_html": "<p>Let <span class=\"katex--inline\">\\mathcal{P}</span> be the image when the parabola <span class=\"katex--inline\">y=x^2</span> is rotated <span class=\"katex--inline\">45^\\circ</span> clockwise around the origin. Then, the equation <span class=\"katex--inline\">x^2+y^2-\\sqrt{2}x-\\sqrt{2}y=k</span> intersects <span class=\"katex--inline\">\\mathcal{P}</span> at more than <span class=\"katex--inline\">2</span> points. The values of <span class=\"katex--inline\">k</span> that satisfy this form a half-open interval with length <span class=\"katex--inline\">\\ell</span>. What is <span class=\"katex--inline\">\\ell</span>?</p>&#10;<p><span class=\"katex--inline\">\\textbf{(A)}~\\tfrac{1}{4}\\qquad\\textbf{(B)}~\\tfrac{1}{2}\\qquad\\textbf{(C)}~1\\qquad\\textbf{(D)}~2\\qquad\\textbf{(E)}~4</span></p>&#10;", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 5, "problem_name": "2024 Mock AMC 12 - Problem 24", "can_next": true, "can_prev": true, "nxt": "/problem/2024_mock_amc12-p25", "prev": "/problem/2024_mock_amc12-p23"}}