MathDB

25

Part of 2019 AMC 10

Problems(2)

Factorial Powers Divisibility

Source: AMC 10A #25/AMC 12A #24

2/8/2019
For how many integers nn between 11 and 5050, inclusive, is (n2−1)!(n!)n \frac{(n^2-1)!}{(n!)^n} an integer? (Recall that 0!=10! = 1.)
<spanclass=′latex−bold′>(A)</span>31<spanclass=′latex−bold′>(B)</span>32<spanclass=′latex−bold′>(C)</span>33<spanclass=′latex−bold′>(D)</span>34<spanclass=′latex−bold′>(E)</span>35<span class='latex-bold'>(A) </span> 31 \qquad <span class='latex-bold'>(B) </span> 32 \qquad <span class='latex-bold'>(C) </span> 33 \qquad <span class='latex-bold'>(D) </span> 34 \qquad <span class='latex-bold'>(E) </span> 35
factorialAMCAMC 102019 AMC2019 AMC 12A2019 AMC 10AAMC 12
Clever Recursion

Source: 2019 AMC 10B #25/2019 AMC 12B #23

2/14/2019
How many sequences of 00s and 11s of length 1919 are there that begin with a 00, end with a 00, contain no two consecutive 00s, and contain no three consecutive 11s?
<spanclass=′latex−bold′>(A)</span>55<spanclass=′latex−bold′>(B)</span>60<spanclass=′latex−bold′>(C)</span>65<spanclass=′latex−bold′>(D)</span>70<spanclass=′latex−bold′>(E)</span>75<span class='latex-bold'>(A) </span>55\qquad<span class='latex-bold'>(B) </span>60\qquad<span class='latex-bold'>(C) </span>65\qquad<span class='latex-bold'>(D) </span>70\qquad<span class='latex-bold'>(E) </span>75
AMCAMC 10AMC 10 BAMC 12AMC 12 Bcool solution rusczyk