MathDB

p1. The serpent of fire and the serpent of ice play a game. Since the serpent of ice loves the lucky number

6

, he will roll a fair

6

-sided die with faces numbered

1

through

6

. The serpent of fire will pay him

\log_{10} x

, where

x

is the number he rolls. The serpent of ice rolls the die

6

times. His expected total amount of winnings across the

6

rounds is

k

. Find

10^k

.
p2. Let

a = \log_3 5

b = \log_3 4

c = - \log_3 20

, evaluate

\frac{a^2+b^2}{a^2+b^2+ab} +\frac{b^2+c^2}{b^2+c^2+bc} +\frac{c^2+a^2}{c^2+a^2+ca}

.
p3. Let

\vartriangle ABC

be an isosceles obtuse triangle with

AB = AC

and circumcenter

O

. The circle with diameter

AO

meets

BC

at points

X, Y

, where X is closer to

B

. Suppose

XB = Y C = 4

XY = 6

, and the area of

\vartriangle ABC

m\sqrt{n}

for positive integers

m

and

n

, where

n

does not contain any square factors. Find

m + n

.
p4. Alice is not sure what to have for dinner, so she uses a fair

6

-sided die to decide. She keeps rolling, and if she gets all the even numbers (i.e. getting all of

2, 4, 6

) before getting any odd number, she will reward herself with McDonald’s. Find the probability that Alice could have McDonald’s for dinner.
p5. How many distinct ways are there to split

50

apples,

50

oranges,

50

bananas into two boxes, such that the products of the number of apples, oranges, and bananas in each box are nonzero and equal?
p6. Sujay and Rishabh are taking turns marking lattice points within a square board in the Cartesian plane with opposite vertices

(1, 1)

(n, n)

for some constant

n

. Sujay loses when the two-point pattern

P

below shows up:https://cdn.artofproblemsolving.com/attachments/1/9/d1fe285294d4146afc0c7a2180b15586b04643.png That is, Sujay loses when there exists a pair of points

(x, y)

and

(x + 2, y + 1)

. He and Rishabh stop marking points when the pattern

P

appears on the board. If Rishabh goes first, let

S

be the set of all integers

3 \le n \le 100

such that Rishabh has a strategy to always trick Sujay into being the one who creates

P

. Find the sum of all elements of

S

.
p7. Let

a

be the shortest distance between the origin

(0, 0)

and the graph of

y^3 = x(6y -x^2)-8

. Find

\lfloor a^2 \rfloor

. (

\lfloor x\rfloor

is the largest integer not exceeding

x

)
p8. Find all real solutions to the following equation:

2\sqrt2x^2 + x -\sqrt{1 - x^2 } -\sqrt2 = 0.

p9. Given the expression

S = (x^4 - x)(x^2 - x^3)

for

x = \cos \frac{2\pi}{5 }+ i\sin \frac{2\pi}{5 }

, find the value of

S^2

.
p10. In a

32

team single-elimination rock-paper-scissors tournament, the teams are numbered from

1

32

. Each team is guaranteed (through incredible rock-paper-scissors skill) to win any match against a team with a higher number than it, and therefore will lose to any team with a lower number. Each round, teams who have not lost yet are randomly paired with other teams, and the losers of each match are eliminated. After the

5

rounds of the tournament, the team that won all

5

rounds is ranked

1

st, the team that lost the 5th round is ranked

2

nd, and the two teams that lost the

4

th round play each other for

3

rd and

4

th place. What is the probability that the teams numbered

1, 2, 3

, and

4

are ranked

1

st, 2nd, 3rd, and 4th respectively? If the probability is

\frac{m}{n}

for relatively prime integers

m

and

n

, find

m

.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

Problem Statement