MathDB

Round 1
p1. A small pizza costs

\$4

and has

6

slices. A large pizza costs

\$9

and has

14

slices. If the MMATHS organizers got at least

400

slices of pizza (having extra is okay) as cheaply as possible, how many large pizzas did they buy?
p2. Rachel flips a fair coin until she gets a tails. What is the probability that she gets an even number of heads before the tails?
p3. Find the unique positive integer

n

that satisfies

n! \cdot (n + 1)! = (n + 4)!

.
Round 2
p4. The Portland Malt Shoppe stocks

10

ice cream flavors and

8

mix-ins. A milkshake consists of exactly

1

flavor of ice cream and between

1

and

3

mix-ins. (Mix-ins can be repeated, the number of each mix-in matters, and the order of the mix-ins doesn’t matter.) How many different milkshakes can be ordered?
p5. Find the minimum possible value of the expression

(x)^2 + (x + 3)^4 + (x + 4)^4 + (x + 7)^2

, where

x

is a real number.
p6. Ralph has a cylinder with height

15

and volume

\frac{960}{\pi}

. What is the longest distance (staying on the surface) between two points of the cylinder?
Round 3
p7. If there are exactly

3

pairs

(x, y)

satisfying

x^2 + y^2 = 8

and

x + y = (x - y)^2 + a

, what is the value of

a

?
p8. If

n

is an integer between

4

and

1000

, what is the largest possible power of

2

that

n^4 - 13n^2 + 36

could be divisible by? (Your answer should be this power of

2

, not just the exponent.)
p9. Find the sum of all positive integers

n \ge 2

for which the following statement is true: “for any arrangement of

n

points in three-dimensional space where the points are not all collinear, you can always find one of the points such that the

n - 1

rays from this point through the other points are all distinct.”
Round 4
p10. Donald writes the number

12121213131415

on a piece of paper. How many ways can he rearrange these fourteen digits to make another number where the digit in every place value is different from what was there before?
p11. A question on Joe’s math test asked him to compute

\frac{a}{b} +\frac34

, where

a

and

b

were both integers. Because he didn’t know how to add fractions, he submitted

\frac{a+3}{b+4}

as his answer. But it turns out that he was right for these particular values of

a

and

b

! What is the largest possible value that a could have been?
p12. Christopher has a globe with radius

r

inches. He puts his finger on a point on the equator. He moves his finger

5\pi

inches North, then

\pi

inches East, then

5\pi

inches South, then

2\pi

inches West. If he ended where he started, what is the largest possible value of

r

?
PS. You should use hide for answers. Rounds 5-7 have be posted [url=https://artofproblemsolving.com/community/c4h2789002p24519497]here. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

Problem Statement