{"status": "success", "data": {"description_md": "Consider functions $f$ that satisfy $$|f(x)-f(y)|\\leq \\frac{1}{2}|x-y|$$ for all real numbers $x$ and $y$. Of all such functions that also satisfy the equation $f(300) = f(900)$, what is the greatest possible value of\n$$f(f(800))-f(f(400))?$$\n\n$\\textbf{(A)}\\ 25 \\qquad\\textbf{(B)}\\ 50 \\qquad\\textbf{(C)}\\ 100 \\qquad\\textbf{(D)}\\ 150 \\qquad\\textbf{(E)}\\ 200$", "description_html": "<p>Consider functions  <span class=\"katex--inline\">f</span>  that satisfy  <span class=\"katex--display\">|f(x)-f(y)|\\leq \\frac{1}{2}|x-y|</span>  for all real numbers  <span class=\"katex--inline\">x</span>  and  <span class=\"katex--inline\">y</span> . Of all such functions that also satisfy the equation  <span class=\"katex--inline\">f(300) = f(900)</span> , what is the greatest possible value of<br/>\n <span class=\"katex--display\">f(f(800))-f(f(400))?</span> </p>\n<p> <span class=\"katex--inline\">\\textbf{(A)}\\ 25 \\qquad\\textbf{(B)}\\ 50 \\qquad\\textbf{(C)}\\ 100 \\qquad\\textbf{(D)}\\ 150 \\qquad\\textbf{(E)}\\ 200</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": 4, "problem_name": "2022 AMC 10B Problem 24", "can_next": true, "can_prev": true, "nxt": "/problem/22_amc10B_p25", "prev": "/problem/22_amc10B_p23"}}