MathDB

17

Part of 2023 AMC 12/AHSME

Problems(2)

a new addition to the frog family

Source: 2023 AMC12A #17

11/9/2023
Flora the frog starts at 00 on the number line and makes a sequence of jumps to the right. In any one jump, independent of previous jumps, Flora leaps a positive integer distance mm with probability 12m\frac{1}{2^m}. What is the probability that Flora will eventually land at 1010?
<spanclass=′latex−bold′>(A)</span>5512<spanclass=′latex−bold′>(B)</span>451024<spanclass=′latex−bold′>(C)</span>1271024<spanclass=′latex−bold′>(D)</span>5111024<spanclass=′latex−bold′>(E)</span>12<span class='latex-bold'>(A) </span> \frac{5}{512} \qquad <span class='latex-bold'>(B) </span> \frac{45}{1024} \qquad <span class='latex-bold'>(C) </span> \frac{127}{1024} \qquad <span class='latex-bold'>(D) </span> \frac{511}{1024} \qquad <span class='latex-bold'>(E) </span> \frac{1}{2}
probabilityAMCAMC 122023 AMC2023 AMC 12A
The yearly LoC

Source: 2023 AMC 12B #17

11/15/2023
Triangle ABCABC has side lengths in arithmetic progression, and the smallest side has length 6.6. If the triangle has an angle of 120∘,120^\circ, what is the area of ABCABC?
<spanclass=′latex−bold′>(A)</span> 123<spanclass=′latex−bold′>(B)</span> 86<spanclass=′latex−bold′>(C)</span> 142<spanclass=′latex−bold′>(D)</span> 202<spanclass=′latex−bold′>(E)</span> 153<span class='latex-bold'>(A)</span>\ 12\sqrt 3 \qquad<span class='latex-bold'>(B)</span>\ 8\sqrt 6 \qquad<span class='latex-bold'>(C)</span>\ 14\sqrt 2 \qquad<span class='latex-bold'>(D)</span>\ 20\sqrt 2 \qquad<span class='latex-bold'>(E)</span>\ 15\sqrt 3
AMCAMC 12geometryarithmetic sequence2023 AMC2023 AMC 12B