MathDB

Round 1
p1. This year, the Mathathon consists of

7

rounds, each with

3

problems. Another math test, Aspartaime, consists of

3

rounds, each with

5

problems. How many more problems are on the Mathathon than on Aspartaime?
p2. Let the solutions to

x^3 + 7x^2 - 242x - 2016 = 0

a, b

, and

c

. Find

a^2 + b^2 + c^2

. (You might find it helpful to know that the roots are all rational.)
p3. For triangle

ABC

, you are given

AB = 8

and

\angle A = 30^o

. You are told that

BC

will be chosen from amongst the integers from

1

10

, inclusive, each with equal probability. What is the probability that once the side length

BC

is chosen there is exactly one possible triangle

ABC

?
Round 2
p4. It’s raining! You want to keep your cat warm and dry, so you want to put socks, rain boots, and plastic bags on your cat’s four paws. Note that for each paw, you must put the sock on before the boot, and the boot before the plastic bag. Also, the items on one paw do not affect the items you can put on another paw. How many different orders are there for you to put all twelve items of rain footwear on your cat?
p5. Let

a

be the square root of the least positive multiple of

2016

that is a square. Let

b

be the cube root of the least positive multiple of

2016

that is a cube. What is

a - b

?
p6. Hypersomnia Cookies sells cookies in boxes of

6, 9

10

. You can only buy cookies in whole boxes. What is the largest number of cookies you cannot exactly buy? (For example, you couldn’t buy

8

cookies.)
Round 3
p7. There is a store that sells each of the

26

letters. All letters of the same type cost the same amount (i.e. any ‘a’ costs the same as any other ‘a’), but different letters may or may not cost different amounts. For example, the cost of spelling “trade” is the same as the cost of spelling “tread,” even though the cost of using a ‘t’ may be different from the cost of an ‘r.’ If the letters to spell out

1

cost

\$1001

, the letters to spell out

2

cost

\$1010

, and the letters to spell out

11

cost

\$2015

, how much do the letters to spell out

12

cost?
p8. There is a square

ABCD

with a point

P

inside. Given that

PA = 6

PB = 9

PC = 8

. Calculate

PD

.
p9. How many ordered pairs of positive integers

(x, y)

are solutions to

x^2 - y^2 = 2016

?
Round 4
p10. Given a triangle with side lengths

5, 6

and

7

, calculate the sum of the three heights of the triangle.
p11. There are

6

people in a room. Each person simultaneously points at a random person in the room that is not him/herself. What is the probability that each person is pointing at someone who is pointing back to them?
p12. Find all

x

such that

\sum_{i=0}^{\infty} ix^i =\frac34

.
PS. You should use hide for answers. Rounds 5-7 have been posted [url=https://artofproblemsolving.com/community/c4h2782837p24446063]here. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

Problem Statement