{"status": "success", "data": {"description_md": "Let $@$ denote the \"averaged with\" operation: $a @ b = (a+b)/2$. Which of the following distributive laws hold for all numbers $x, y,$ and $z$? $$\\text{I. x @ (y + z) = (x @ y) + (x @ z)}$$ $$\\text{II. x + (y @ z) = (x + y) @ (x + z)}$$ $$\\text{III. x @ (y @ z) = (x @ y) @ (x @ z)}$$\n\n$\\textbf{(A)}\\ \\text{I only} \\qquad\\textbf{(B)}\\ \\text{II only} \\qquad\\textbf{(C)}\\ \\text{III only} \\qquad\\textbf{(D)}\\ \\text{I and III only} \\qquad\\textbf{(E)}\\ \\text{II and III only}$", "description_html": "<p>Let  <span class=\"katex--inline\">@</span>  denote the &#8220;averaged with&#8221; operation:  <span class=\"katex--inline\">a @ b = (a+b)/2</span> . Which of the following distributive laws hold for all numbers  <span class=\"katex--inline\">x, y,</span>  and  <span class=\"katex--inline\">z</span> ?  <span class=\"katex--display\">\\text{I. x @ (y + z) = (x @ y) + (x @ z)}</span>   <span class=\"katex--display\">\\text{II. x + (y @ z) = (x + y) @ (x + z)}</span>   <span class=\"katex--display\">\\text{III. x @ (y @ z) = (x @ y) @ (x @ z)}</span> </p>\n<p> <span class=\"katex--inline\">\\textbf{(A)}\\ \\text{I only} \\qquad\\textbf{(B)}\\ \\text{II only} \\qquad\\textbf{(C)}\\ \\text{III only} \\qquad\\textbf{(D)}\\ \\text{I and III only} \\qquad\\textbf{(E)}\\ \\text{II and III only}</span> </p>\n<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": "2011 AMC 10B Problem 15", "can_next": true, "can_prev": true, "nxt": "/problem/11_amc10B_p16", "prev": "/problem/11_amc10B_p14"}}