MathDB

p1. Find

20 \cdot 18 + 20 + 18 + 1

.
p2. Suzie’s Ice Cream has

10

flavors of ice cream,

5

types of cones, and

5

toppings to choose from. An ice cream cone consists of one flavor, one cone, and one topping. How many ways are there for Sebastian to order an ice cream cone from Suzie’s?
p3. Let

a = 7

and

b = 77

. Find

\frac{(2ab)^2}{(a+b)^2-(a-b)^2}

.
p4. Sebastian invests

100,000

dollars. On the first day, the value of his investment falls by

20

percent. On the second day, it increases by

25

percent. On the third day, it falls by

25

percent. On the fourth day, it increases by

60

percent. How many dollars is his investment worth by the end of the fourth day?
p5. Square

ABCD

has side length

5

. Points

K,L,M,N

are on segments

AB

BC

CD

DA

respectively,such that

MC = CL = 2

and

NA = AK = 1

. The area of trapezoid

KLMN

can be expressed as

\frac{m}{n}

for relatively prime positive integers

m

and

n

. Find

m + n

.
p6. Suppose that

p

and

q

are prime numbers. If

p + q = 30

, find the sum of all possible values of

pq

.
p7. Tori receives a

15 - 20 - 25

right triangle. She cuts the triangle into two pieces along the altitude to the side of length

25

. What is the difference between the areas of the two pieces?
p8. The factorial of a positive integer

n

, denoted

n!

, is the product of all the positive integers less than or equal to

n

. For example,

1! = 1

and

5! = 120

. Let

m!

and

n!

be the smallest and largest factorial ending in exactly

3

zeroes, respectively. Find

m + n

.
p9. Sam is late to class, which is located at point

B

. He begins his walk at point

A

and is only allowed to walk on the grid lines. He wants to get to his destination quickly; how many paths are there that minimize his walking distance? https://cdn.artofproblemsolving.com/attachments/a/5/764e64ac315c950367357a1a8658b08abd635b.png
p10. Mr. Iyer owns a set of

6

antique marbles, where

1

is red,

2

are yellow, and

3

are blue. Unfortunately, he has randomly lost two of the marbles. His granddaughter starts drawing the remaining

4

out of a bag without replacement. She draws a yellow marble, then the red marble. Suppose that the probability that the next marble she draws is blue is equal to

\frac{m}{n}

, where

m

and

n

are relatively prime positiveintegers. What is

m + n

?
p11. If

a

is a positive integer, what is the largest integer that will always be a factor of

(a^3+1)(a^3+2)(a^3+3)

?
p12. What is the largest prime number that is a factor of

160,401

?
p13. For how many integers

m

does the equation

x^2 + mx + 2018 = 0

have no real solutions in

x

?
p14. What is the largest palindrome that can be expressed as the product of two two-digit numbers? A palindrome is a positive integer that has the same value when its digits are reversed. An example of a palindrome is

7887887

.
p15. In circle

\omega

inscribe quadrilateral

ADBC

such that

AB \perp CD

. Let

E

be the intersection of diagonals

AB

and

CD

, and suppose that

EC = 3

ED = 4

, and

EB = 2

. If the radius of

\omega

r

, then

r^2 =\frac{m}{n}

for relatively prime positive integers

m

and

n

. Determine

m + n

.
p16. Suppose that

a, b, c

are nonzero real numbers such that

2a^2 + 5b^2 + 45c^2 = 4ab + 6bc + 12ca

. Find the value of

\frac{9(a + b + c)^3}{5abc}

.
p17. Call a positive integer n spicy if there exist n distinct integers

k_1, k_2, ... , k_n

such that the following two conditions hold:

\bullet

|k_1| + |k_2| +... + |k_n| = n2

\bullet

k_1 + k_2 + ...+ k_n = 0

. Determine the number of spicy integers less than

10^6

.
p18. Consider the system of equations

|x^2 - y^2 - 4x + 4y| = 4

|x^2 + y^2 - 4x - 4y| = 4.

Find the sum of all

x

and

y

that satisfy the system.
p19. Determine the number of

8

letter sequences, consisting only of the letters

W,Q,N

, in which none of the sequences

WW

QQQ

, or

NNNN

appear. For example,

WQQNNNQQ

is a valid sequence, while

WWWQNQNQ

is not.
p20. Triangle

\vartriangle ABC

has

AB = 7

CA = 8

, and

BC = 9

. Let the reflections of

A,B,C

over the orthocenter H be

A'

B'

C'

. The area of the intersection of triangles

ABC

and

A'B'C'

can be expressed in the form

\frac{a\sqrt{b}}{c}

, where

b

is squarefree and

a

and

c

are relatively prime. determine

a+b+c

. (The orthocenter of a triangle is the intersection of its three altitudes.)
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

Problem Statement