MathDB

2022 Alg/NT Div 2 P5

Source:

2/28/2022

Alan is assigning values to lattice points on the 3d coordinate plane. First, Alan computes the roots of the cubic

20x^3-22x^2+2x+1

and finds that they are

\alpha

,

\beta

, and

\gamma

. He finds out that each of these roots satisfy

|\alpha|,|\beta|,|\gamma|\leq 1

On each point

(x,y,z)

where

x,y,

and

z

are all nonnegative integers, Alan writes down

\alpha^x\beta^y\gamma^z

. What is the value of the sum of all numbers he writes down?
Proposed by Alan Abraham

algebranumber theory

2022 Geo Div 2 P5

Source:

2/28/2022

Let

ABC

be an equilateral triangle of unit side length and suppose

D

is a point on segment

\overline{BC}

such that

DB<DC.

Let

M

and

N

denote the midpoints of

\overline{AB}

and

\overline{AC},

respectively. Suppose

X

and

Y

are the intersections of lines

AB

and

ND,

and lines

AC

and

MD,

respectively. Given that

XY=4,

what is the value of

\frac{DB}{DC}?

Proposed by Kyle Lee

geometry

2022 Combo Div 2 P5

Source:

2/28/2022

Daniel, Ethan, and Zack are playing a multi-round game of Tetris. Whoever wins

11

rounds first is crowned the champion. However Zack is trying to pull off a "reverse-sweep", where (at-least) one of the other two players first hits

10

wins while Zack is still at

0

, but Zack still ends up being the first to reach

11

. How many possible sequences of round wins can lead to Zack pulling off a reverse sweep?
Proposed by Dilhan Salgado

combinatorics

2.5