MathDB

p1. Five friends arrive at a hotel which has three rooms. Rooms

A

and

B

hold two people each, and room

C

holds one person. How many different ways could the five friends lodge for the night?
p2. The set of numbers

\{1, 3, 8, 12, x\}

has the same average and median. What is the sum of all possible values of

x

? (Note that

x

is not necessarily greater than

12

.)
p3. How many four-digit numbers

\overline{ABCD}

are there such that the three-digit number

\overline{BCD}

satisfies

\overline{BCD} = \frac16 \overline{ABCD}

? (Note that

A

must be nonzero.)
p4. Find the smallest positive integer

n

such that

n

leaves a remainder of

5

when divided by

14

n^2

leaves a remainder of

1

when divided by

12

, and

n^3

leaves a remainder of

7

when divided by

10

.
p5. In rectangle

ABCD

, let

E

lie on

\overline{CD}

, and let

F

be the intersection of

\overline{AC}

and

\overline{BE}

. If the area of

\vartriangle ABF

45

and the area of

\vartriangle CEF

20

, find the area of the quadrilateral

ADEF

.
p6. If

x

and

y

are integers and

14x^2y^3 - 38x^2 + 21y^3 = 2018

, what is the value of

x^2y

?
p7.

A, B, C, D

all lie on a circle with

\overline{AB} = \overline{BC} = \overline{CD}

. If the distance between any two of these points is a positive integer, what is the smallest possible perimeter of quadrilateral

ABCD

?
p8. Compute

\sum^{\infty}_{m=1} \sum^{\infty}_{n=1} \frac{m\cos^2(n) + n \sin^2(m)}{3^{m+n}(m + n)}

p9. Diane has a collection of weighted coins with different probabilities of landing on heads, and she flips nine coins sequentially according to a particular set of rules. She uses a coin that always lands on heads for her first and second flips, and she uses a coin that always lands on tails for her third flip. For each subsequent flip, she chooses a coin to flip as follows: if she has so far flipped

a

heads out of

b

total flips, then she uses a coin with an

\frac{a}{b}

probability of landing on heads. What is the probability that after all nine flips, she has gotten six heads and three tails?
p10. For any prime number

p

, let

S_p

be the sum of all the positive divisors of

37^pp^{37}

(including

1

and

37^pp^{37}

). Find the sum of all primes

p

such that

S_p

is divisible by

p

.
p11. Six people are playing poker. At the beginning of the game, they have

1

2

3

4

5

, and

6

dollars, respectively. At the end of the game, nobody has lost more than a dollar, and each player has a distinct nonnegative integer dollar amount. (The total amount of money in the game remains constant.) How many distinct finishing rankings (i.e. lists of first place through sixth place) are possible?
p12. Let

C_1

be a circle of radius

1

, and let

C_2

be a circle of radius

\frac12

internally tangent to

C_1

. Let

\{\omega_0, \omega_1, ... \}

be an infinite sequence of circles, such that

\omega_0

has radius

\frac12

and each

\omega_k

is internally tangent to

C_1

and externally tangent to both

C_2

and

\omega_{k+1}

. (The

\omega_k

’s are mutually distinct.) What is the radius of

\omega_{100}

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

Problem Statement