MathDB

20

Part of 2018 Purple Comet Problems

Problems(2)

Purple Comet 2018 MS problem 20

Source:

3/19/2020
Let ABCDABCD be a square with side length 66. Circles X,YX, Y , and ZZ are congruent circles with centers inside the square such that XX is tangent to both sides AB‾\overline{AB} and AD‾\overline{AD}, YY is tangent to both sides AB‾\overline{AB} and BC‾\overline{BC}, and ZZ is tangent to side CD‾\overline{CD} and both circles XX and YY . The radius of the circle XX can be written m−nm -\sqrt{n}, where mm and nn are positive integers. Find m+nm + n.
geometry
Purple Comet 2018 HS problem 20

Source:

3/19/2020
Aileen plays badminton where she and her opponent stand on opposite sides of a net and attempt to bat a birdie back and forth over the net. A player wins a point if their opponent fails to bat the birdie over the net. When Aileen is the server (the first player to try to hit the birdie over the net), she wins a point with probability 910\frac{9}{10} . Each time Aileen successfully bats the birdie over the net, her opponent, independent of all previous hits, returns the birdie with probability 34\frac{3}{4} . Each time Aileen bats the birdie, independent of all previous hits, she returns the birdie with probability mn\frac{m}{n} , where mm and nn are relatively prime positive integers. Find m+nm + n.
probability