MathDB

21

Part of 2019 AMC 10

Problems(2)

Triangle in Space With Each of Its Sides Tangent to a Sphere

Source: 2019 AMC 10A #21 / 2019 AMC 12A #18

2/8/2019
A sphere with center OO has radius 66. A triangle with sides of length 15,15,15, 15, and 2424 is situated in space so that each of its sides is tangent to the sphere. What is the distance between OO and the plane determined by the triangle?
<spanclass=′latex−bold′>(A)</span>23<spanclass=′latex−bold′>(B)</span>4<spanclass=′latex−bold′>(C)</span>32<spanclass=′latex−bold′>(D)</span>25<spanclass=′latex−bold′>(E)</span>5 <span class='latex-bold'>(A) </span>2\sqrt{3}\qquad <span class='latex-bold'>(B) </span>4\qquad <span class='latex-bold'>(C) </span>3\sqrt{2}\qquad <span class='latex-bold'>(D) </span>2\sqrt{5}\qquad <span class='latex-bold'>(E) </span>5\qquad
geometryAMC 10AMCAMC 122019 AMC 12A2019 AMC 10A2019 AMC
Debra's Coin

Source: 2019 AMC 10 B #21

2/14/2019
Debra flips a fair coin repeatedly, keeping track of how many heads and how many tails she has seen in total, until she gets either two heads in a row or two tails in a row, at which point she stops flipping. What is the probability that she gets two heads in a row but she sees a second tail before she sees a second head?
<spanclass=′latex−bold′>(A)</span>136<spanclass=′latex−bold′>(B)</span>124<spanclass=′latex−bold′>(C)</span>118<spanclass=′latex−bold′>(D)</span>112<spanclass=′latex−bold′>(E)</span>16<span class='latex-bold'>(A) </span> \frac{1}{36} \qquad <span class='latex-bold'>(B) </span> \frac{1}{24} \qquad <span class='latex-bold'>(C) </span> \frac{1}{18} \qquad <span class='latex-bold'>(D) </span> \frac{1}{12} \qquad <span class='latex-bold'>(E) </span> \frac{1}{6}
2019 AMCAMCAMC 102019 AMC 10Bprobability