{"status": "success", "data": {"description_md": "A cone is filled with water. Two solid spheres are placed in the cone as shown in the diagram and water spills out. (The spheres are touching each other, each sphere touches the cone all of the way around, and the top of the top sphere is level with the top of the cone.) The larger sphere has radius twice that of the smaller sphere. If the volume of the water remaining in the cone is $2016\\pi$, what is the radius of the smaller sphere? (The volume of a sphere with radius $r$ is $\\frac{4}{3}\\pi r^3$. The volume of a cone with radius $r$ and height $h$ is $\\frac{1}{3}\\pi r^2 h$.)\n\n<center><img class=\"problem-image\" src=\"/static/cayley/2013/crops/13_cayley_p24_fig.png\" alt=\"figure\" width=\"135\"></center>\n\n$\\text{(A)}\\ 2\\sqrt{2}\\qquad\\text{(B)}\\ 6\\qquad\\text{(C)}\\ 8\\qquad\\text{(D)}\\ 6\\sqrt{2}\\qquad\\text{(E)}\\ 4\\sqrt[3]{2}$\n\n___\n\n**[Cayley](https://www.cemc.uwaterloo.ca/contests/past_contests.html#pcf), by [CEMC](https://cemc.uwaterloo.ca/), is licensed under the [CC BY-NC 3.0 DEED](https://creativecommons.org/licenses/by-nc/3.0/). They are not affiliated with or endorse TopsOJ in any way.**\n", "description_html": "<p>A cone is filled with water. Two solid spheres are placed in the cone as shown in the diagram and water spills out. (The spheres are touching each other, each sphere touches the cone all of the way around, and the top of the top sphere is level with the top of the cone.) The larger sphere has radius twice that of the smaller sphere. If the volume of the water remaining in the cone is <span class=\"katex--inline\">2016\\pi</span>, what is the radius of the smaller sphere? (The volume of a sphere with radius <span class=\"katex--inline\">r</span> is <span class=\"katex--inline\">\\frac{4}{3}\\pi r^3</span>. The volume of a cone with radius <span class=\"katex--inline\">r</span> and height <span class=\"katex--inline\">h</span> is <span class=\"katex--inline\">\\frac{1}{3}\\pi r^2 h</span>.)</p>\n<center><img class=\"problem-image\" src=\"/static/cayley/2013/crops/13_cayley_p24_fig.png\" alt=\"figure\" width=\"135\"></center>\n<p><span class=\"katex--inline\">\\text{(A)}\\ 2\\sqrt{2}\\qquad\\text{(B)}\\ 6\\qquad\\text{(C)}\\ 8\\qquad\\text{(D)}\\ 6\\sqrt{2}\\qquad\\text{(E)}\\ 4\\sqrt[3]{2}</span></p>\n<hr>\n<p><strong><a href=\"https://www.cemc.uwaterloo.ca/contests/past_contests.html#pcf\">Cayley</a>, by <a href=\"https://cemc.uwaterloo.ca/\">CEMC</a>, is licensed under the <a href=\"https://creativecommons.org/licenses/by-nc/3.0/\">CC BY-NC 3.0 DEED</a>. They are not affiliated with or endorse TopsOJ in any way.</strong></p>\n", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 3, "problem_name": "2013 Cayley Problem 24", "can_next": true, "can_prev": true, "nxt": "/problem/13_cayley_p25", "prev": "/problem/13_cayley_p23"}}