MathDB

22

Part of 2020 BMT Fall

Problems(2)

2020 BMT Team 22

Source:

1/9/2022
Suppose that x,yx, y, and zz are positive real numbers satisfying {x2+xy+y2=64y2+yz+z2=49z2+zx+x2=57\begin{cases} x^2 + xy + y^2 = 64 \\ y^2 + yz + z^2 = 49 \\ z^2 + zx + x^2 = 57 \end{cases} Then xyz3\sqrt[3]{xyz} can be expressed as m/nm/n , where mm and nn are relatively prime positive integers. Compute m+nm + n.
algebrasystem of equations
2020 BMT Individual 22

Source:

1/9/2022
Three lights are placed horizontally on a line on the ceiling. All the lights are initially off. Every second, Neil picks one of the three lights uniformly at random to switch: if it is off, he switches it on; if it is on, he switches it off. When a light is switched, any lights directly to the left or right of that light also get turned on (if they were off) or off (if they were on). The expected number of lights that are on after Neil has flipped switches three times can be expressed in the form m/nm/ n , where mm and nn are relatively prime positive integers. Compute m+nm + n.
combinatorics