MathDB

23

Part of 2021 AMC 10 Fall

Problems(2)

Divisor Function

Source: 2021 AMC10A #23, 2021 AMC12A #20

11/11/2021
For each positive integer nn, let f1(n)f_1(n) be twice the number of positive integer divisors of nn, and for j≥2j \ge 2, let fj(n)=f1(fj−1(n))f_j(n) = f_1(f_{j-1}(n)). For how many values of n≤50n \le 50 is f50(n)=12?f_{50}(n) = 12?
<spanclass=′latex−bold′>(A)</span>7<spanclass=′latex−bold′>(B)</span>8<spanclass=′latex−bold′>(C)</span>9<spanclass=′latex−bold′>(D)</span>10<spanclass=′latex−bold′>(E)</span>11<span class='latex-bold'>(A) </span>7\qquad<span class='latex-bold'>(B) </span>8\qquad<span class='latex-bold'>(C) </span>9\qquad<span class='latex-bold'>(D) </span>10\qquad<span class='latex-bold'>(E) </span>11
function
Take the AoPS AMC practice test

Source: 2021 Fall AMC 10B #23

11/17/2021
Each of the 55{ } sides and the 55{ } diagonals of a regular pentagon are randomly and independently colored red or blue with equal probability. What is the probability that there will be a triangle whose vertices are among the vertices of the pentagon such that all of its sides have the same color?
(<spanclass=′latex−bold′>A</span>) 23(<spanclass=′latex−bold′>B</span>) 105128(<spanclass=′latex−bold′>C</span>) 125128(<spanclass=′latex−bold′>D</span>) 253256(<spanclass=′latex−bold′>E</span>) 1(<span class='latex-bold'>A</span>)\: \frac23\qquad(<span class='latex-bold'>B</span>) \: \frac{105}{128}\qquad(<span class='latex-bold'>C</span>) \: \frac{125}{128}\qquad(<span class='latex-bold'>D</span>) \: \frac{253}{256}\qquad(<span class='latex-bold'>E</span>) \: 1
AMC 10AMCAMC 10 B