MathDB

2.8 1.4

Part of 2022 CMIMC

Problems(3)

2022 Alg/NT Div 1 P4 (Div 2 P8)

Source:

2/28/2022
Let zz be a complex number that satisfies the equation z−4z2−5z+1+2z−42z2−5z+1+z−2z2−3z+1=3z.\frac{z-4}{z^2-5z+1} + \frac{2z-4}{2z^2-5z+1} + \frac{z-2}{z^2-3z+1} = \frac{3}{z}. Over all possible values of zz, find the sum of the values of ∣1z2−5z+1+12z2−5z+1+1z2−3z+1∣.\left| \frac{1}{z^2-5z+1} + \frac{1}{2z^2-5z+1} + \frac{1}{z^2-3z+1} \right|.
Proposed by Justin Hsieh
algebranumber theory
2022 Geo Div 1 P4 (Div 2 P8)

Source:

2/28/2022
Let AA and BB be points on circle Γ\Gamma such that AB=10.AB=\sqrt{10}. Point CC is outside Γ\Gamma such that △ABC\triangle ABC is equilateral. Let DD be a point on Γ\Gamma and suppose the line through CC and DD intersects ABAB and Γ\Gamma again at points EE and F≠D.F \neq D. It is given that points C,D,E,FC, D, E, F are collinear in that order and that CD=DE=EF.CD=DE=EF. What is the area of Γ?\Gamma?
Proposed by Kyle Lee
geometry
2022 Combo Div 1 P4 (Div 2 P8)

Source:

2/28/2022
The CMU Kiltie Band is attempting to crash a helicopter via grappling hook. The helicopter starts parallel (angle 00 degrees) to the ground. Each time the band members pull the hook, they tilt the helicopter forward by either xx or x+1x+1 degrees, with equal probability, if the helicopter is currently at an angle xx degrees with the ground. Causing the helicopter to tilt to 9090 degrees or beyond will crash the helicopter. Find the expected number of times the band must pull the hook in order to crash the helicopter.
Proposed by Justin Hsieh
combinatorics