MathDB

7

Part of 2017 AMC 12/AHSME

Problems(2)

Recursive Function

Source: AMC 12A Problem 7

2/8/2017
Define a function on the positive integers recursively by f(1)=2f(1) = 2, f(n)=f(n−1)+1f(n) = f(n-1) + 1 if nn is even, and f(n)=f(n−2)+2f(n) = f(n-2) + 2 if nn is odd and greater than 11. What is f(2017)f(2017)?
<spanclass=′latex−bold′>(A)</span>2017<spanclass=′latex−bold′>(B)</span>2018<spanclass=′latex−bold′>(C)</span>4034<spanclass=′latex−bold′>(D)</span>4035<spanclass=′latex−bold′>(E)</span>4036<span class='latex-bold'>(A) </span> 2017 \qquad <span class='latex-bold'>(B) </span> 2018 \qquad <span class='latex-bold'>(C) </span> 4034 \qquad <span class='latex-bold'>(D) </span> 4035 \qquad <span class='latex-bold'>(E) </span> 4036
AMCAMC 12function2017 AMC 12A
Period of Trigonometric Function

Source: 2017 AMC 12B #7

2/16/2017
The functions sin⁡(x)\sin(x) and cos⁡(x)\cos(x) are periodic with least period 2π2\pi. What is the least period of the function cos⁡(sin⁡(x))\cos(\sin(x))?
<spanclass=′latex−bold′>(A)</span> π2<spanclass=′latex−bold′>(B)</span> π<spanclass=′latex−bold′>(C)</span> 2π<spanclass=′latex−bold′>(D)</span> 4π<spanclass=′latex−bold′>(E)</span><span class='latex-bold'>(A)</span>\ \frac{\pi}{2}\qquad<span class='latex-bold'>(B)</span>\ \pi\qquad<span class='latex-bold'>(C)</span>\ 2\pi\qquad<span class='latex-bold'>(D)</span>\ 4\pi\qquad<span class='latex-bold'>(E)</span> It's not periodic.
AMCAMC 12AMC 12 Bfunction2017 AMC 12B