MathDB

Distinct Factor Product

Source:

8/8/2024

Let

2\leq N\leq 2022

be a positive integer. Find the sum of all possible values of

N

such that the product of the distinct divisors of

N

is

N^{\frac{21}{2}}

.

number theory2022

Angle Bisector Intersection and Area

Source:

8/8/2024

Triangle

WSE

has side lengths

WS=13

,

SE=15

, and

WE=14

. Points

J

and

H

lie on

\overline{SE}

such that

SJ=JH=HE=5

. Let the angle bisector of

\angle{WES}

intersect

\overline{WH}

and

\overline{WJ}

at points

M

and

T

, respectively. Find the area of quadrilateral

JHMT

.

geometryangle bisector2022

Two Minima Conditions

Source:

8/9/2024

Let

x

be a variable that can take any positive real value. For certain positive real constants

s

and

t

, the value of

x^2 + \frac{s}{x}

is minimized at

x = t

, and the value of

t^2\ln(2 + tx) + \frac{1}{x^2}

is minimized at

x = s

. Compute the ordered pair

(s, t)

.

calculusoptimization2022

Drawing Marbles

Source:

8/8/2024

Dr. G has a bag of five marbles and enjoys drawing one marble from the bag, uniformly at random, and then putting it back in the bag. How many draws, on average, will it take Dr. G to reach a point where every marble has been drawn at least once?

expected value2022

Game Theory Optimal Betting

Source:

8/8/2024

Andy, Bella, and Chris are playing in a trivia contest. Andy has

21,200

points, Bella has

23,600

points, and Chris has

11,200

points. They have reached the final round, which works as follows:
[*] they are given a hint as to what the only question of the round will be about; [*] then, each of them must bet some amount of their points—this bet must be a nonnegative integer (a player does not know any of the other players' bets, and this bet cannot be changed later on); [*] then, they will be shown the question, where they will have

30

seconds to individually submit a response (a player does not know any of the other players' answers); [*] finally, once time runs out, their responses and bets are revealed—if a player's response is correct, then the number of points they bet will be added to their score, otherwise, it will be subtracted from their score, and whoever ends up having the most points wins the contest (if there is a tie for the win, then the winner is randomly decided).
Suppose that the contestants are currently deciding their bets based on the hint that the question will be about history. Bella knows that she will likely get the question wrong, but she also knows that Andy, who dislikes history, will definitely get it wrong. Knowing this, Bella wagers an amount that will guarantee her a win. Find the maximum number of points Bella could have ended up with.

optimization2022Game Theory

3

Problems(5)