MathDB

14

Part of 2017 AMC 10

Problems(2)

Soda is not healthy

Source: 2017 AMC10 #14

2/8/2017
Every week Roger pays for a movie ticket and a soda out of his allowance. Last week, Roger's allowance was AA dollars. The cost of his movie ticket was 20%20\% of the difference between AA and the cost of his soda, while the cost of his soda was 5%5\% of the difference between AA and the cost of his movie ticket. To the nearest whole percent, what fraction of AA did Roger pay for his movie ticket and soda?
<spanclass=′latex−bold′>(A)</span>9%<spanclass=′latex−bold′>(B)</span>19%<spanclass=′latex−bold′>(C)</span>22%<spanclass=′latex−bold′>(D)</span>23%<spanclass=′latex−bold′>(E)</span>25% <span class='latex-bold'>(A) </span>9\%\qquad <span class='latex-bold'>(B) </span> 19\%\qquad <span class='latex-bold'>(C) </span> 22\%\qquad <span class='latex-bold'>(D) </span> 23\%\qquad <span class='latex-bold'>(E) </span>25\%
AMCAMC 10percent2017 AMC 10A
Fermat's Little Theorem

Source: 2017 AMC 10B #14

2/16/2017
An integer NN is selected at random in the range 1≤N≤2020.1\le N \le 2020. What is the probability that the remainder when N16N^{16} is divided by 55 is 11?
<spanclass=′latex−bold′>(A)</span> 15<spanclass=′latex−bold′>(B)</span> 25<spanclass=′latex−bold′>(C)</span> 35<spanclass=′latex−bold′>(D)</span> 45<spanclass=′latex−bold′>(E)</span> 1<span class='latex-bold'>(A)</span> \text{ }\frac{1}{5} \qquad <span class='latex-bold'>(B)</span> \text{ }\frac{2}{5} \qquad <span class='latex-bold'>(C)</span> \text{ }\frac{3}{5} \qquad <span class='latex-bold'>(D)</span> \text{ }\frac{4}{5} \qquad <span class='latex-bold'>(E)</span> \text{ 1}
AMCAMC 10AMC 10 Bnumber theory2017 AMC 10B