MathDB

Set 4
p16. Albert, Beatrice, Corey, and Dora are playing a card game with two decks of cards numbered

1-50

each. Albert, Beatrice, and Corey draw cards from the same deck without replacement, but Dora draws from the other deck. What is the probability that the value of Corey’s card is the highest value or is tied for the highest value of all

4

drawn cards?
p17. Suppose that

s

is the sum of all positive values of

x

that satisfy

2016\{x\} = x+[x]

. Find

\{s\}

. (Note:

[x]

denotes the greatest integer less than or equal to

x

and

\{x\}

denotes

x - [x]

.)
p18. Let

ABC

be a triangle such that

AB = 41

BC = 52

, and

CA = 15

. Let H be the intersection of the

B

altitude and

C

altitude. Furthermore let

P

be a point on

AH

. Both

P

and

H

are reflected over

BC

to form

P'

and

H'

. If the area of triangle

P'H'C

60

, compute

PH

.
p19. A random integer

n

is chosen between

1

and

30

, inclusive. Then, a random positive divisor of

n, k

, is chosen. What is the probability that

k^2 > n

?
p20. What are the last two digits of the value

3^{361}

?
Set 5
p21. Let

f(n)

denote the number of ways a

3 \times n

board can be completely tiled with

1 \times 3

and

1 \times 4

tiles, without overlap or any tiles hanging over the edge. The tiles may be rotated. Find

\sum^9_{i=0} f(i) = f(0) + f(1) + ... + f(8) + f(9)

. By convention,

f(0) = 1

.
p22. Find the sum of all

5

-digit perfect squares whose digits are all distinct and come from the set

\{0, 2, 3, 5, 7, 8\}

.
p23. Mary is flipping a fair coin. On average, how many flips would it take for Mary to get

4

heads and

2

tails?
p24. A cylinder is formed by taking the unit circle on the

xy

-plane and extruding it to positive infinity. A plane with equation

z = 1 - x

truncates the cylinder. As a result, there are three surfaces: a surface along the lateral side of the cylinder, an ellipse formed by the intersection of the plane and the cylinder, and the unit circle. What is the total surface area of the ellipse formed and the lateral surface? (The area of an ellipse with semi-major axis

a

and semi-minor axis

b

\pi ab

.)
p25. Let the Blair numbers be defined as follows:

B_0 = 5

B_1 = 1

, and

B_n = B_{n-1} + B_{n-2}

for all

n \ge 2

. Evaluate

\sum_{i=0}^{\infty} \frac{B_i}{51^i}= B_0 +\frac{B_1}{51} +\frac{B_2}{51^2} +\frac{B_3}{51^3} +...

Estimation
p26. Choose an integer between

1

and

10

, inclusive. Your score will be the number you choose divided by the number of teams that chose your number.
p27.

2016

blind people each bring a hat to a party and leave their hat in a pile at the front door. As each partier leaves, they take a random hat from the ones remaining in a pile. Estimate the probability that at least

1

person gets their own hat back.
p28. Estimate how many lattice points lie within the graph of

|x^3| + |y^3| < 2016

.
p29. Consider all ordered pairs of integers

(x, y)

with

1 \le x, y \le 2016

. Estimate how many such ordered pairs are relatively prime.
p30. Estimate how many times the letter “e” appears among all Guts Round questions.
PS. You should use hide for answers. First sets have been posted [url=https://artofproblemsolving.com/community/c3h2779594p24402189]here. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

Problem Statement