2022 AMC 10 B

Complete problem set with solutions and individual problem pages

Problem 24 Hard

Consider functions f that satisfy |f(x)-f(y)|\leqslant\dfrac{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

f\left( f(800)\right)-f(f(400))?

  • A.

    25

  • B.

    50

  • C.

    100

  • D.

    150

  • E.

    200

Answer:B

|f(800)-f(900) |\leqslant \frac{1}{2} |800-900|=50;

|f(400)-f(300)| \leqslant \frac{1}{2} |400-300|=50.

|f(800)-f(400)| = |f(800-f(900)+f(300)-f(400)|

\leqslant |f(800)-f(900)|+|f(300)-f(400)|

\leqslant 50+50 =100.

\begin{aligned} f[f(800)]-f[f(400)] & \leqslant \frac{1}{2}|f(800)-f(400)| \\ & \leqslant \frac{1}{2} \times 100 \\ & =40\end{aligned}

Related Problems
Problem 21 2022 AMC 10 B
Problem 22 2022 AMC 10 B
Problem 23 2022 AMC 10 B
Problem 25 2022 AMC 10 B
Think Academy Powered by ChatGPT