MathDB

24

Part of 2022 AMC 10

Problems(2)

bijection moment

Source: 2022 AMC 10/12A #24

11/11/2022
How many strings of length 55 formed from the digits 00,11,22,33,44 are there such that for each j∈{1,2,3,4}j\in\{1,2,3,4\}, at least jj of the digits are less than jj? (For example, 0221402214 satisfies the condition because it contains at least 11 digit less than 11, at least 22 digits less than 22, at least 33 digits less than 33, and at least 44 digits less than 44. The string 2340423404 does not satisfy the condition because it does not contain at least 22 digits less than 22.)
<spanclass=′latex−bold′>(A)</span>500<spanclass=′latex−bold′>(B)</span>625<spanclass=′latex−bold′>(C)</span>1089<spanclass=′latex−bold′>(D)</span>1199<spanclass=′latex−bold′>(E)</span>1296<span class='latex-bold'>(A) </span>500\qquad<span class='latex-bold'>(B) </span>625\qquad<span class='latex-bold'>(C) </span>1089\qquad<span class='latex-bold'>(D) </span>1199\qquad<span class='latex-bold'>(E) </span>1296
AMCAMC 102022 AMC 12A2022 AMC2022 AMC 10aAMC 12
AMC is functionally equating again

Source: 2022 AMC10B P24

11/17/2022
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
algebrafunctional equationAMCAMC 102022 AMC2022 AMC 10B