MathDB

Round 5
p13. Pieck the Frog hops on Pascal’s Triangle, where she starts at the number

1

at the top. In a hop, Pieck can hop to one of the two numbers directly below the number she is currently on with equal probability. Given that the expected value of the number she is on after

7

hops is

\frac{m}{n}

, where

m

and

n

are relatively prime positive integers, find

m+n

.
p14. Maisy chooses a random set

(x, y)

that satisfies

x^2 + y^2 -26x -10y \le 482.

The probability that

y>0

can be expressed as

\frac{A\pi -B\sqrt{C}}{D \pi}

. Find

A+B +C +D

.
[color=#f00]Due to the problem having a typo, all teams who inputted answers received points
p15.

6

points are located on a circle. How many ways are there to draw any number of line segments between the points such that none of the line segments overlap and none of the points are on more than one line segment? (It is possible to draw no line segments).
Round 6
p16. Find the number of

3

3

grids such that each square in the grid is colored white or black and no two black squares share an edge.
p17. Let

ABC

be a triangle with side lengths

AB = 20

BC = 25

, and

AC = 15

. Let

D

be the point on BC such that

CD = 4

. Let

E

be the foot of the altitude from

A

BC

. Let

F

be the intersection of

AE

with the circle of radius

7

centered at

A

such that

F

is outside of triangle

ABC

DF

can be expressed as

\sqrt{m}

, where

m

is a positive integer. Find

m

.
p18. Bill and Frank were arrested under suspicion for committing a crime and face the classic Prisoner’s Dilemma. They are both given the choice whether to rat out the other and walk away, leaving their partner to face a

9

year prison sentence. Given that neither of them talk, they both face a

3

year sentence. If both of them talk, they both will serve a

6

year sentence. Both Bill and Frank talk or do not talk with the same probabilities. Given the probability that at least one of them talks is

\frac{11}{36}

, find the expected duration of Bill’s sentence in months.
Round 7
p19. Rectangle

ABCD

has point

E

on side

\overline{CD}

. Point

F

is the intersection of

\overline{AC}

and

\overline{BE}

. Given that the area of

\vartriangle AFB

175

and the area of

\vartriangle CFE

28

, find the area of

ADEF

.
p20. Real numbers

x, y

, and

z

satisfy the system of equations

5x+ 13y -z = 100,

25x^2 +169y^2 -z2 +130x y= 16000,

80x +208y-2z = 2020.

Find the value of

x yz

.
[color=#f00]Due to the problem having infinitely many solutions, all teams who inputted answers received points.
p21. Bob is standing at the number

1

on the number line. If Bob is standing at the number

n

, he can move to

n +1

n +2

, or

n +4

. In howmany different ways can he move to the number

10

?
Round 8
p22. A sequence

a_1,a_2,a_3, ...

of positive integers is defined such that

a_1 = 4

, and for each integer

k \ge 2

2(a_{k-1} +a_k +a_{k+1}) = a_ka_{k-1} +8.

Given that

a_6 = 488

, find

a_2 +a_3 +a_4 +a_5

.
p23.

\overline{PQ}

is a diameter of circle

\omega

with radius

1

and center

O

. Let

A

be a point such that

AP

is tangent to

\omega

. Let

\gamma

be a circle with diameter

AP

. Let

A'

be where

AQ

hits the circle with diameter

AP

and

A''

be where

AO

hits the circle with diameter

OP

. Let

A'A''

hit

PQ

R

. Given that the value of the length

RA'

is is always less than

k

and

k

is minimized, find the greatest integer less than or equal to

1000k

.
p24. You have cards numbered

1,2,3, ... ,100

all in a line, in that order. You may swap any two adjacent cards at any time. Given that you make

{100 \choose 2}

total swaps, where you swap each distinct pair of cards exactly once, and do not do any swaps simultaneously, find the total number of distinct possible final orderings of the cards.
PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h3166472p28814057]here and 9-12 [url=https://artofproblemsolving.com/community/c3h3166480p28814155]here. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

Problem Statement