MathDB

22

Part of 2023 AMC 12/AHSME

Problems(2)

Integer Divisor Function

Source: 2023 AMC 12A #22

11/9/2023
Let ff be the unique function defined on the positive integers such that ∑d∣nd⋅f(nd)=1\sum_{d\mid n}d\cdot f\left(\frac{n}{d}\right)=1 for all positive integers nn, where the sum is taken over all positive divisors of nn. What is f(2023)f(2023)?
<spanclass=′latex−bold′>(A)</span> −1536<spanclass=′latex−bold′>(B)</span> 96<spanclass=′latex−bold′>(C)</span> 108<spanclass=′latex−bold′>(D)</span> 116<spanclass=′latex−bold′>(E)</span> 144<span class='latex-bold'>(A)</span>~-1536\qquad<span class='latex-bold'>(B)</span>~96\qquad<span class='latex-bold'>(C)</span>~108\qquad<span class='latex-bold'>(D)</span>~116\qquad<span class='latex-bold'>(E)</span>~144
AMCAMC 12AMC 12A2023 AMC2023 AMC 12Afunctions
funtcion

Source: 2023 AMC 12B #22

11/15/2023
A real-valued function ff has the property that for all real numbers aa and b,b, f(a+b)+f(a−b)=2f(a)f(b).f(a + b) + f(a - b) = 2f(a) f(b). Which one of the following cannot be the value of f(1)?f(1)? <spanclass=′latex−bold′>(A)</span>0<spanclass=′latex−bold′>(B)</span>1<spanclass=′latex−bold′>(C)</span>−1<spanclass=′latex−bold′>(D)</span>2<spanclass=′latex−bold′>(E)</span>−2 <span class='latex-bold'>(A) </span> 0 \qquad <span class='latex-bold'>(B) </span> 1 \qquad <span class='latex-bold'>(C) </span> -1 \qquad <span class='latex-bold'>(D) </span> 2 \qquad <span class='latex-bold'>(E) </span> -2
function2023 AMC2023 AMC 12BAMCAMC 12