MathDB

6

Part of 2019 CMIMC

Problems(4)

2019 A/NT6: The Obligatory Vieta Problem

Source:

1/27/2019
Let a,ba, b and cc be the distinct solutions to the equation x3−2x2+3x−4=0x^3-2x^2+3x-4=0. Find the value of 1a(b2+c2−a2)+1b(c2+a2−b2)+1c(a2+b2−c2).\frac{1}{a(b^2+c^2-a^2)}+\frac{1}{b(c^2+a^2-b^2)}+\frac{1}{c(a^2+b^2-c^2)}.
algebrapolynomialVieta2019
2019 C/CS6: Lightbulbs in a Circle

Source:

1/27/2019
There are 100100 lightbulbs B1,…,B100B_1,\ldots, B_{100} spaced evenly around a circle in this order. Additionally, there are 100100 switches S1,…,S100S_1,\ldots, S_{100} such that for all 1≤i≤1001\leq i\leq 100, switch SiS_i toggles the states of lights Bi−1B_{i-1} and Bi+1B_{i+1} (where here B101=B1B_{101} = B_1). Suppose David chooses whether to flick each switch with probability 12\tfrac12. What is the expected number of lightbulbs which are on at the end of this process given that not all lightbulbs are off?
2019combinatorics
2019 G6: Parallel to Side of Triangle Through Arc Midpoint

Source:

1/27/2019
Let ABCABC be a triangle with AB=209AB=209, AC=243AC=243, and ∠BAC=60∘\angle BAC = 60^\circ, and denote by NN the midpoint of the major arc BAC^\widehat{BAC} of circle ⊙(ABC)\odot(ABC). Suppose the parallel to ABAB through NN intersects BC‾\overline{BC} at a point XX. Compute the ratio BXXC\tfrac{BX}{XC}.
2019geometry
2019 T6: Minimum Value of Sum of Sines

Source:

1/27/2019
Across all x∈Rx \in \mathbb{R}, find the maximum value of the expression sin⁡x+sin⁡3x+sin⁡5x.\sin x + \sin 3x + \sin 5x.
2019teaminequalities