MathDB

19

Part of 2021 AMC 10 Fall

Problems(2)

Disk Rolling

Source: 2022 AMC 10A #19

11/11/2021
A disk of radius 11 rolls all the way around the inside of a square of side length s>4s>4 and sweeps out a region of area AA. A second disk of radius 11 rolls all the way around the outside of the same square and sweeps out a region of area 2A2A. The value of ss can be written as a+bπca+\frac{b\pi}{c}, where a,ba,b, and cc are positive integers and bb and cc are relatively prime. What is a+b+ca+b+c?
<spanclass=′latex−bold′>(A)</span> 10<spanclass=′latex−bold′>(B)</span> 11<spanclass=′latex−bold′>(C)</span> 12<spanclass=′latex−bold′>(D)</span> 13<spanclass=′latex−bold′>(E)</span> 14<span class='latex-bold'>(A)</span> ~10\qquad<span class='latex-bold'>(B)</span> ~11\qquad<span class='latex-bold'>(C)</span> ~12\qquad<span class='latex-bold'>(D)</span> ~13\qquad<span class='latex-bold'>(E)</span> ~14
AMCAMC 10AMC 10 A
Frivolous Functions

Source: 2021 AMC 10B #19

11/17/2021
Let NN be the positive integer 7777…7777777\ldots777, a 313313-digit number where each digit is a 77. Let f(r)f(r) be the leading digit of the rr{ }th root of NN. What isf(2)+f(3)+f(4)+f(5)+f(6)?f(2) + f(3) + f(4) + f(5)+ f(6)? (<spanclass=′latex−bold′>A</span>) 8(<spanclass=′latex−bold′>B</span>) 9(<spanclass=′latex−bold′>C</span>) 11(<spanclass=′latex−bold′>D</span>) 22(<spanclass=′latex−bold′>E</span>) 29(<span class='latex-bold'>A</span>)\: 8\qquad(<span class='latex-bold'>B</span>) \: 9\qquad(<span class='latex-bold'>C</span>) \: 11\qquad(<span class='latex-bold'>D</span>) \: 22\qquad(<span class='latex-bold'>E</span>) \: 29
functionAMCAMC 10AMC 10 B