MathDB

p1. What is the smallest possible value for the product of two real numbers that differ by ten?
p2. Determine the number of positive integers

n

with

1 \le n \le 400

that satisfy the following:

\bullet

n

is a square number.

\bullet

n

is one more than a multiple of

5

\bullet

n

is even.
p3. How many positive integers less than

2019

are either a perfect cube or a perfect square but not both?
p4. Felicia draws the heart-shaped figure

GOAT

that is made of two semicircles of equal area and an equilateral triangle, as shown below. If

GO = 2

, what is the area of the figure? https://cdn.artofproblemsolving.com/attachments/3/c/388daa657351100f408ab3f1185f9ab32fcca5.png
p5. For distinct digits

A, B

, and

C

: \begin{tabular}{cccc} & A & A \\ & B & B \\ + & C & C \\ \hline A & B & C \\ \end{tabular} Compute

A \cdot B \cdot C

.
p6 What is the difference between the largest and smallest value for

lcm(a,b,c)

, where

a,b

, and

c

are distinct positive integers between

1

and

10

, inclusive?
p7. Let

A

and

B

be points on the circumference of a circle with center

O

such that

\angle AOB = 100^o

. If

X

is the midpoint of minor arc

AB

and

Y

is on the circumference of the circle such that

XY\perp AO

, find the measure of

\angle OBY

.
p8. When Ben works at twice his normal rate and Sammy works at his normal rate, they can finish a project together in

6

hours. When Ben works at his normal rate and Sammy works as three times his normal rate, they can finish the same project together in

4

hours. How many hours does it take Ben and Sammy to finish that project if they each work together at their normal rates?
p9. How many positive integer divisors

n

20000

are there such that when

20000

is divided by

n

, the quotient is divisible by a square number greater than

1

?
p10. What’s the maximum number of Friday the

13

th’s that can occur in a year?
p11. Let circle

\omega

pass through points

B

and

C

of triangle

ABC

. Suppose

\omega

intersects segment

AB

at a point

D \ne B

and intersects segment

AC

at a point

E \ne C

. If

AD = DC = 12

DB = 3

, and

EC = 8

, determine the length of

EB

.
p12. Let

a,b

be integers that satisfy the equation

2a^2 - b^2 + ab = 18

. Find the ordered pair

(a,b)

.
p13. Let

a,b,c

be nonzero complex numbers such that

a -\frac{1}{b}= 8, b -\frac{1}{c}= 10, c -\frac{1}{a}= 12.

Find

abc -\frac{1}{abc}

.
p14. Let

\vartriangle ABC

be an equilateral triangle of side length

1

. Let

\omega_0

be the incircle of

\vartriangle ABC

, and for

n > 0

, define the infinite progression of circles

\omega_n

as follows:

\bullet

\omega_n

is tangent to

AB

and

AC

and externally tangent to

\omega_{n-1}

\bullet

The area of

\omega_n

is strictly less than the area of

\omega_{n-1}

. Determine the total area enclosed by all

\omega_i

for

i \ge 0

.
p15. Determine the remainder when

13^{2020} +11^{2020}

is divided by

144

.
p16. Let

x

be a solution to

x +\frac{1}{x}= 1

. Compute

x^{2019} +\frac{1}{x^{2019}}

.
p17. The positive integers are colored black and white such that if

n

is one color, then

2n

is the other color. If all of the odd numbers are colored black, then how many numbers between

100

and

200

inclusive are colored white?
p18. What is the expected number of rolls it will take to get all six values of a six-sided die face-up at least once?
p19. Let

\vartriangle ABC

have side lengths

AB = 19

BC = 2019

, and

AC = 2020

. Let

D,E

be the feet of the angle bisectors drawn from

A

and

B

, and let

X,Y

to be the feet of the altitudes from

C

AD

and

C

BE

, respectively. Determine the length of

XY

.
p20. Suppose I have

5

unit cubes of cheese that I want to divide evenly amongst

3

hungry mice. I can cut the cheese into smaller blocks, but cannot combine blocks into a bigger block. Over all possible choices of cuts in the cheese, what’s the largest possible volume of the smallest block of cheese?
PS. You had better use hide for answers.

Problem Statement