MathDB

9

Part of 2015 ASDAN Math Tournament

Problems(5)

2015 Advanced #9

Source:

7/8/2022
You play a game with a biased coin, which has probability 34\tfrac{3}{4} of landing heads. Each time you toss heads, you score 11 point, while tossing tails earns no points. After any turn, you can stop playing the game and keep the points you currently have. However, if you are still playing when you toss tails for the second time, you lose all of your points. If you play to maximize your expected score, what is your expected score from playing this game?
2015Advanced Topics Test
2015 Algebra #9

Source:

7/1/2022
Compute all pairs of nonzero real numbers (x,y)(x,y) such that xx2+y+yx+y2=−1and1x+1y=1.\frac{x}{x^2+y}+\frac{y}{x+y^2}=-1\qquad\text{and}\qquad\frac{1}{x}+\frac{1}{y}=1.
2015Algebra Test
2015 Guts #9

Source:

7/28/2022
Compute the sum of the digits of 1016101^6.
2015Guts Test
2015 Geometry #9

Source:

7/1/2022
Regular tetrahedron ABCDABCD has center OO and side length 11. Points A′A', B′B', C′C', and D′D' are defined by reflecting AA, BB, CC, and DD about OO. Compute the volume of the polyhedron with vertices ABCDA′B′C′D′ABCDA'B'C'D'.
2015Geometry Test
2015 Team #9

Source:

8/3/2022
A coin is tossed 1010 times. Compute the probability that two heads will turn up in succession somewhere in the sequence of throws.
2015team test