MathDB

p1. Nikhil computes the sum of the first

10

positive integers, starting from

1

. He then divides that sum by 5. What remainder does he get?
p2. In class, starting at

8:00

, Ava claps her hands once every

4

minutes, while Ella claps her hands once every

6

minutes. What is the smallest number of minutes after

8:00

such that both Ava and Ella clap their hands at the same time?
p3. A triangle has side lengths

3

4

, and

5

. If all of the side lengths of the triangle are doubled, how many times larger is the area?
p4. There are

50

students in a room. Every student is wearing either

0

1

, or

2

shoes. An even number of the students are wearing exactly

1

shoe. Of the remaining students, exactly half of them have

2

shoes and half of them have

0

shoes. How many shoes are worn in total by the

50

students?
p5. What is the value of

-2 + 4 - 6 + 8 - ... + 8088

?
p6. Suppose Lauren has

2

cats and

2

dogs. If she chooses

2

of the

4

pets uniformly at random, what is the probability that the 2 chosen pets are either both cats or both dogs?
p7. Let triangle

\vartriangle ABC

be equilateral with side length

6

. Points

E

and

F

lie on

BC

such that

E

is closer to

B

than it is to

C

and

F

is closer to

C

than it is to

B

. If

BE = EF = FC

, what is the area of triangle

\vartriangle AFE

?
p8. The two equations

x^2 + ax - 4 = 0

and

x^2 - 4x + a = 0

share exactly one common solution for

x

. Compute the value of

a

.
p9. At Shreymart, Shreyas sells apples at a price

c

. A customer who buys

n

apples pays

nc

dollars, rounded to the nearest integer, where we always round up if the cost ends in

.5

. For example, if the cost of the apples is

4.2

dollars, a customer pays

4

dollars. Similarly, if the cost of the apples is

4.5

dollars, a customer pays

5

dollars. If Justin buys

7

apples for

3

dollars and

4

apples for

1

dollar, how many dollars should he pay for

20

apples?
p10. In triangle

\vartriangle ABC

, the angle trisector of

\angle BAC

closer to

\overline{AC}

than

\overline{AB}

intersects

\overline{BC}

D

. Given that triangle

\vartriangle ABD

is equilateral with area

1

, compute the area of triangle

\vartriangle ABC

.
p11. Wanda lists out all the primes less than

100

for which the last digit of that prime equals the last digit of that prime's square. For instance,

71

is in Wanda's list because its square,

5041

, also has

1

as its last digit. What is the product of the last digits of all the primes in Wanda's list?
p12. How many ways are there to arrange the letters of

SUSBUS

such that

SUS

appears as a contiguous substring? For example,

SUSBUS

and

USSUSB

are both valid arrangements, but

SUBSSU

is not.
p13. Suppose that

x

and

y

are integers such that

x \ge 5

y \ge 3

, and

\sqrt{x - 5} +\sqrt{y - 3} = \sqrt{x + y}

. Compute the maximum possible value of

xy

.
p14. What is the largest integer

k

divisible by

14

such that

x^2-100x+k = 0

has two distinct integer roots?
p15. What is the sum of the first

16

positive integers whose digits consist of only

0

s and

1

s?
p16. Jonathan and Ajit are flipping two unfair coins. Jonathan's coin lands on heads with probability

\frac{1}{20}

while Ajit's coin lands on heads with probability

\frac{1}{22}

. Each year, they flip their coins at thesame time, independently of their previous flips. Compute the probability that Jonathan's coin lands on heads strictly before Ajit's coin does.
p17. A point is chosen uniformly at random in square

ABCD

. What is the probability that it is closer to one of the

4

sides than to one of the

2

diagonals?
p18. Two integers are coprime if they share no common positive factors other than

1

. For example,

3

and

5

are coprime because their only common factor is

1

. Compute the sum of all positive integers that are coprime to

198

and less than

198

.
p19. Sumith lists out the positive integer factors of

12

in a line, writing them out in increasing order as

1

2

3

4

6

12

. Luke, being the mischievious person he is, writes down a permutation of those factors and lists it right under Sumith's as

a_1

a_2

a_3

a_4

a_5

a_6

. Luke then calculates

gcd(a_1, 2a_2, 3a_3, 4a_4, 6a_5, 12a_6).

Given that Luke's result is greater than

1

, how many possible permutations could he have written?
p20. Tetrahedron

ABCD

is drawn such that

DA = DB = DC = 2

\angle ADB = \angle ADC = 90^o

, and

\angle BDC = 120^o

. Compute the radius of the sphere that passes through

A

B

C

, and

D

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

Problem Statement