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AMC is functionally equating again

Source: 2022 AMC10B P24

November 17, 2022
algebrafunctional equationAMCAMC 102022 AMC2022 AMC 10B

Problem Statement

Consider functions ff that satisfy ∣f(x)−f(y)∣≤12∣x−y∣|f(x)-f(y)|\leq \frac{1}{2}|x-y| for all real numbers xx and yy. Of all such functions that also satisfy the equation f(300)=f(900)f(300) = f(900), what is the greatest possible value of
f(f(800))−f(f(400))?f(f(800))-f(f(400))?
<spanclass=′latex−bold′>(A)</span> 25<spanclass=′latex−bold′>(B)</span> 50<spanclass=′latex−bold′>(C)</span> 100<spanclass=′latex−bold′>(D)</span> 150<spanclass=′latex−bold′>(E)</span> 200 <span class='latex-bold'>(A)</span>\ 25 \qquad <span class='latex-bold'>(B)</span>\ 50 \qquad <span class='latex-bold'>(C)</span>\ 100 \qquad <span class='latex-bold'>(D)</span>\ 150 \qquad <span class='latex-bold'>(E)</span>\ 200
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