MathDB

p1. Evaluate

(2-0)^2 \cdot 3+ \frac{20}{2+3}

.
p2. Let

x = 11 \cdot 99

and

y = 9 \cdot 101

. Find the sumof the digits of

x \cdot y

.
p3. A rectangle is cut into two pieces. The ratio between the areas of the two pieces is

3 : 1

and the positive difference between those areas is

20

. What’s the area of the rectangle?
p4. Edgeworth is scared of elevators. He is currently on floor

50

of a building, and he wants to go down to floor

1

. Edgeworth can go down at most

4

floors each time he uses the elevator. What’s the minimum number of times he needs to use the elevator to get to floor

1

?
p5. There are

20

people at a party. Fifteen of those people are normal and

5

are crazy. A normal person will shake hands once with every other normal person, while a crazy person will shake hands twice with every other crazy person. How many total handshakes occur at the party?
p6. Wam and Sang are chewing gum. Gum comes in packages, each package consisting of

14

sticks of gum. Wam eats

6

packs and

9

individual sticks of gum. Sang wants to eat twice as much gum as Wam. How many packs of gum must Sang buy?
p7. At Lakeside Health School (LHS),

40\%

of students are male and

60\%

of the students are female. If half of the students at the school take biology, and the same number ofmale and female students take biology, to the nearest percent, what percent of female students take biology?
p8. Evin is bringing diluted raspberry iced tea to the annual LexingtonMath Team party. He has a cup with

10

mL of iced tea and a

2000

mL cup of water with

10\%

raspberry iced tea. If he fills up the cup with

20

more mL of

10\%

raspberry iced tea water, what percent of the solution will be iced tea?
p9. Tree

1

starts at height

220

m and grows continuously at

3

m per year. Tree

2

starts at height

20

m and grows at

5

m during the first year,

7

m per during the second year,

9

m during the third year, and in general

(3+2n)

m in the nth year. After which year is Tree

2

taller than Tree

1

?
p10. Leo and Chris are playing a game in which Chris flips a coin. The coin lands on heads with probability

\frac{499}{999}

, tails with probability

\frac{499}{999}

, and it lands on its side with probability

\frac{1}{999}

. For each flip of the coin, Leo agrees to give Chris

4

dollars if it lands on heads, nothing if it lands on tails, and

2

dollars if it lands on its side. What’s the expected value of the number of dollars Chris gets after flipping the coin

17

times?
p11. Ephram has a pile of balls, which he tries to divide into piles. If he divides the balls into piles of

7

, there are

5

balls that don’t get divided into any pile. If he divides the balls into piles of

11

, there are

9

balls that aren’t in any pile. If he divides the balls into piles of

13

, there are

11

balls that aren’t in any pile. What is the minimumnumber of balls Ephram has?
p12. Let

\vartriangle ABC

be a triangle such that

AB = 3

BC = 4

, and

C A = 5

. Let

F

be the midpoint of

AB

. Let

E

be the point on

AC

such that

EF \parallel BC

. Let CF and

BE

intersect at

D

. Find

AD

.
p13. Compute the sum of all even positive integers

n \le 1000

such that:

lcm(1,2, 3, ..., (n -1)) \ne lcm(1,2, 3,, ...,n)

.
p14. Find the sum of all palindromes with

6

digits in binary, including those written with leading zeroes.
p15. What is the side length of the smallest square that can entirely contain

3

non-overlapping unit circles?
p16. Find the sum of the digits in the base

7

representation of

6250000

. Express your answer in base

10

.
p17. A number

n

is called sus if

n^4

is one more than a multiple of

59

. Compute the largest sus number less than

2023

.
p18. Michael chooses real numbers

a

and

b

independently and randomly from

(0, 1)

. Given that

a

and

b

differ by at most

\frac14

, what is the probability

a

and

b

are both greater than

\frac12

?
p19. In quadrilateral

ABCD

AB = 7

and

DA = 5

BC =CD

\angle BAD = 135^o

and

\angle BCD = 45^o

. Find the area of

ABCD

.
p20. Find the value of

\sum_{i |210} \sum_{j |i} \left \lfloor \frac{i +1}{j} \right \rfloor

p21. Let

a_n

be the number of words of length

n

with letters

\{A,B,C,D\}

that contain an odd number of

A

s. Evaluate

a_6

.
p22. Detective Hooa is investigating a case where a criminal stole someone’s pizza. There are

69

people involved in the case, among whom one is the criminal and another is a witness of the crime. Every day, Hooa is allowed to invite any of the people involved in the case to his rather large house for questioning. If on some given day, the witness is present and the criminal is not, the witness will reveal who the criminal is. What is the minimum number of days of questioning required such that Hooa is guaranteed to learn who the criminal is?
p23. Find

\sum^{\infty}_{n=2} \frac{2n +10}{n^3 +4n^2 +n -6}.

p24. Let

\vartriangle ABC

be a triangle with circumcircle

\omega

such that

AB = 1

\angle B = 75^o

, and

BC =\sqrt2

. Let lines

\ell_1

and

\ell_2

be tangent to

\omega

A

and

C

respectively. Let

D

be the intersection of

\ell_1

and

\ell_2

. Find

\angle ABD

(in degrees).
p25. Find the sum of the prime factors of

14^6 +27

.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

Problem Statement