MathDB

Round 1
p1. Jom and Terry both flip a fair coin. What is the probability both coins show the same side?
p2. Under the same standard air pressure, when measured in Fahrenheit, water boils at

212^o

F and freezes at

32^o

F. At thesame standard air pressure, when measured in Delisle, water boils at

0

D and freezes at

150

D. If x is today’s temperature in Fahrenheit and y is today’s temperature expressed in Delisle, we have

y = ax + b

. What is the value of

a + b

? (Ignore units.)
p3. What are the last two digits of

5^1 + 5^2 + 5^3 + · · · + 5^{10} + 5^{11}

?
Round 2
p4. Compute the average of the magnitudes of the solutions to the equation

2x^4 + 6x^3 + 18x^2 + 54x + 162 = 0

.
p5. How many integers between

1

and

1000000

inclusive are both squares and cubes?
p6. Simon has a deck of

48

cards. There are

12

cards of each of the following

4

suits: fire, water, earth, and air. Simon randomly selects one card from the deck, looks at the card, returns the selected card to the deck, and shuffles the deck. He repeats the process until he selects an air card. What is the probability that the process ends without Simon selecting a fire or a water card?
Round 3
p7. Ally, Beth, and Christine are playing soccer, and Ally has the ball. Each player has a decision: to pass the ball to a teammate or to shoot it. When a player has the ball, they have a probability

p

of shooting, and

1 - p

of passing the ball. If they pass the ball, it will go to one of the other two teammates with equal probability. Throughout the game,

p

is constant. Once the ball has been shot, the game is over. What is the maximum value of

p

that makes Christine’s total probability of shooting the ball

\frac{3}{20}

?
p8. If

x

and

y

are real numbers, then what is the minimum possible value of the expression

3x^2 - 12xy + 14y^2

given that

x - y = 3

?
p9. Let

ABC

be an equilateral triangle, let

D

be the reflection of the incenter of triangle

ABC

over segment

AB

, and let

E

be the reflection of the incenter of triangle

ABD

over segment

AD

. Suppose the circumcircle

\Omega

of triangle

ADE

intersects segment

AB

again at

X

. If the length of

AB

1

, find the length of

AX

.
Round 4
p10. Elaine has

c

cats. If she divides

c

5

, she has a remainder of

3

. If she divides

c

7

, she has a remainder of

5

. If she divides

c

9

, she has a remainder of

7

. What is the minimum value

c

can be?
p11. Your friend Donny offers to play one of the following games with you. In the first game, he flips a fair coin and if it is heads, then you win. In the second game, he rolls a

10

-sided die (its faces are numbered from

1

10

)

x

times. If, within those

x

rolls, the number

10

appears, then you win. Assuming that you like winning, what is the highest value of

x

where you would prefer to play the coin-flipping game over the die-rolling game?
p12. Let be the set

X = \{0, 1, 2, ..., 100\}

. A subset of

X

, called

N

, is defined as the set that contains every element

x

X

such that for any positive integer

n

, there exists a positive integer

k

such that n can be expressed in the form

n = x^{a_1}+x^{a_2}+...+x^{a_k}

, for some integers

a_1, a_2, ..., a_k

that satisfy

0 \le a_1 \le a_2 \le ... \le a_k

. What is the sum of the elements in

N

?
PS. You should use hide for answers. Rounds 5-7 have be posted [url=https://artofproblemsolving.com/community/c4h2782880p24446580]here. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

Problem Statement