MathDB

p1. Aidan walks into a skyscraper’s first floor lobby and takes the elevator up

50

floors. After exiting the elevator, he takes the stairs up another

10

floors, then takes the elevator down

30

floors. Find the floor number Aidan is currently on.
p2. Jeff flips a fair coin twice and Kaylee rolls a standard

6

-sided die. The probability that Jeff flips

2

heads and Kaylee rolls a

4

P

. Find

\frac{1}{P}

.
p3. Given that

a\odot b = a + \frac{a}{b}

, find

(4\odot 2)\odot 3

.
p4. The following star is created by gluing together twelve equilateral triangles each of side length

3

. Find the outer perimeter of the star. https://cdn.artofproblemsolving.com/attachments/e/6/ad63edbf93c5b7d4c7e5d68da2b4632099d180.png
p5. In Lexington High School’sMath Team, there are

40

students:

20

of whom do science bowl and

22

of whom who do LexMACS. What is the least possible number of students who do both science bowl and LexMACS?
p6. What is the least positive integer multiple of

3

whose digits consist of only

0

s and

1

s? The number does not need to have both digits.
p7. Two fair

6

-sided dice are rolled. The probability that the product of the numbers rolled is at least

30

can be written as

\frac{a}{b}

where

a

and

b

are relatively prime positive integers. Find

a +b

.
p8. At the LHSMath Team Store,

5

hoodies and

1

jacket cost

\$13

, and

5

jackets and

1

hoodie cost

\$17

. Find how much

15

jackets and

15

hoodies cost, in dollars.
p9. Eric wants to eat ice cream. He can choose between

3

options of spherical ice cream scoops. The first option consists of

4

scoops each with a radius of

3

inches, which costs a total of

\$3

. The second option consists of a scoop with radius

4

inches, which costs a total of

\$2

. The third option consists of

5

scoops each with diameter

2

inches, which costs a total of

\$1

. The greatest possible ratio of volume to cost of ice cream Eric can buy is nπ cubic inches per dollar. Find

3n

.
p10. Jen claims that she has lived during at least part of each of five decades. What is the least possible age that Jen could be? (Assume that age is always rounded down to the nearest integer.)
p11. A positive integer

n

is called Maisylike if and only if

n

has fewer factors than

n -1

. Find the sum of the values of

n

that are Maisylike, between

2

and

10

, inclusive.
p12. When Ginny goes to the nearby boba shop, there is a

30\%

chance that the employee gets her drink order wrong. If the drink she receives is not the one she ordered, there is a

60\%

chance that she will drink it anyways. Given that Ginny drank a milk tea today, the probability she ordered it can be written as

\frac{a}{b}

, where

a

and

b

are relatively prime positive integers. Find the value of

a +b

.
p13. Alex selects an integer

m

between

1

and

100

, inclusive. He notices there are the same number of multiples of

5

as multiples of

7

betweenm and

m+9

, inclusive. Find how many numbers Alex could have picked.
p14. In LMTown there are only rainy and sunny days. If it rains one day there’s a

30\%

chance that it will rain the next day. If it’s sunny one day there’s a

90\%

chance it will be sunny the next day. Over n days, as n approaches infinity, the percentage of rainy days approaches

k\%

. Find

10k

.
p15. A bag of letters contains

3

L’s,

3

M’s, and

3

T’s. Aidan picks three letters at random from the bag with replacement, and Andrew picks three letters at random fromthe bag without replacement. Given that the probability that both Aidan and Andrew pick one each of L, M, and T can be written as

\frac{m}{n}

where

m

and

n

are relatively prime positive integers, find

m+n

.
p16. Circle

\omega

is inscribed in a square with side length

2

. In each corner tangent to

2

of the square’s sides and externally tangent to

\omega

is another circle. The radius of each of the smaller

4

circles can be written as

(a -\sqrt{b})

where

a

and

b

are positive integers. Find

a +b

. https://cdn.artofproblemsolving.com/attachments/d/a/c76a780ac857f745067a8d6c4433efdace2dbb.png
p17. In the nonexistent land of Lexingtopia, there are

10

days in the year, and the Lexingtopian Math Society has

5

members. The probability that no two of the LexingtopianMath Society’s members share the same birthday can be written as

\frac{a}{b}

, where

a

and

b

are relatively prime positive integers. Find

a +b

.
p18. Let

D(n)

be the number of diagonals in a regular

n

-gon. Find

\sum^{26}_{n=3} D(n).

p19. Given a square

A_0B_0C_0D_0

as shown below with side length

1

, for all nonnegative integers

n

, construct points

A_{n+1}

B_{n+1}

C_{n+1}

, and

D_{n+1}

A_nB_n

B_nC_n

C_nD_n

, and

D_nA_n

, respectively, such that

\frac{A_n A_{n+1}}{A_{n+1}B_n}=\frac{B_nB_{n+1}}{B_{n+1}C_n} =\frac{C_nC_{n+1}}{C_{n+1}D_n}=\frac{D_nD_{n+1}}{D_{n+1}A_n} =\frac34.

https://cdn.artofproblemsolving.com/attachments/6/a/56a435787db0efba7ab38e8401cf7b06cd059a.png The sum of the series

\sum^{\infty}_{i=0} [A_iB_iC_iD_i ] = [A_0B_0C_0D_0]+[A_1B_1C_1D_1]+[A_2B_2C_2D_2]...

where

[P]

denotes the area of polygon

P

can be written as

\frac{a}{b}

where

a

and

b

are relatively prime positive integers. Find

a +b

.
p20. Let

m

and

n

be two real numbers such that

\frac{2}{n}+m = 9

\frac{2}{m}+n = 1

Find the sum of all possible values of

m

plus the sumof all possible values of

n

.
p21. Let

\sigma (x)

denote the sum of the positive divisors of

x

. Find the smallest prime

p

such that

\sigma (p!) \ge 20 \cdot \sigma ([p -1]!).

p22. Let

\vartriangle ABC

be an isosceles triangle with

AB = AC

. Let

M

be the midpoint of side

\overline{AB}

. Suppose there exists a point X on the circle passing through points

A

M

, and

C

such that

BMCX

is a parallelogram and

M

and

X

are on opposite sides of line

BC

. Let segments

\overline{AX}

and

\overline{BC}

intersect at a point

Y

. Given that

BY = 8

, find

AY^2

.
p23. Kevin chooses

2

integers between

1

and

100

, inclusive. Every minute, Corey can choose a set of numbers and Kevin will tell him how many of the

2

chosen integers are in the set. How many minutes does Corey need until he is certain of Kevin’s

2

chosen numbers?
p24. Evaluate

1^{-1} \cdot 2^{-1} +2^{-1} \cdot 3^{-1} +3^{-1} \cdot 4^{-1} +...+(2015)^{-1} \cdot (2016)^{-1} \,\,\, (mod \,\,\,2017).

p25. In scalene

\vartriangle ABC

, construct point

D

on the opposite side of

AC

B

such that

\angle ABD = \angle DBC = \angle BC A

and

AD =DC

. Let

I

be the incenter of

\vartriangle ABC

. Given that

BC = 64

and

AD = 225

, find

BI

. https://cdn.artofproblemsolving.com/attachments/b/1/5852dd3eaace79c9da0fd518cfdcd5dc13aecf.png
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

Problem Statement