MathDB

p1. What is the smallest positive prime divisor of

101^2 -99^2

?
p2. The product

81 \cdot 13579

equals

1099A99

for some digit

A

. What is

A

?
p3. On a MMATHS team of

6

students, all students forget which of the

5

MMATHS sites (Yale, UVA, UF, UM, and Columbia) they are supposed to attend. On competition day, each student chooses a random site uniformly and independently of each other to attend, and they each take a flight to the one they picked. What is the probability that all of the sites end up with at least one person from that team?
p4. Square

ABCD

has area

12

. Let

E

be the midpoint of

AD

, and let

F

be the point of intersection of

BE

and

AC

. Find the area of quadrilateral

EFCD

.
p5. Alexander is eating sliced olives for dinner. Olives are sliced into thirds so that each olive is composed of two indistinguishable “end“ pieces and one “middle“ piece. For each olive slice that Alexander puts on his plate, there is a

\frac23

chance that it is an end piece and a

\frac13

chance that it is a middle piece. After placing a randomly selected olive slice on his plate, he checks to see if he can rearrange the slices on his plate so that he has at least one whole olive, which consists of any middle piece and any two end pieces. He can successfully rearrange the olive slices on his plate into at least one whole olive after randomly selecting

n

olive slices with probability of at least

0.8

. What is the minimum n for which this is true?
p6. What is the smallest positive integer

n

whose digits multiply to

8000

?
p7. Mitchell has a bunch of

2\times 1

rectangular tiles. He needs to arrange them in such a way that they exactly cover a

2\times 10

grid of

1\times 1

squares. How many ways are there for him to do this? Two such “tilings“ are considered distinct from each other if not all of the

2

-by-

1

rectangular tiles in the two tilings are placed in exactly the same position and orientation.
p8. Trapezoid

ABCD

is inscribed in a circle of radius

100

. If

AB = 200

and

BC = 50

, what is the perimeter of

ABCD

?
p9. How many ordered pairs of nonzero integers

(a, b)

are there such that

\frac{1}{a} + \frac{1}{b} = \frac{1}{24}

?
p10. Suppose

a

and

b

are complex numbers such that

|a|= |b| = 1

and

a + b = \frac13 + \frac14 i

. Find the product

ab

.
p11. Find the number of ordered triples of positive integers

(a_1, a_2, a_3)

such that

a_1 + a_2 + a_3 = 2017

and

2

does not divide

a_1

3

does not divide

a_2

, and

4

does not divide

a_3

.
p12. Circles

\Gamma

and

\Omega

have radii of

20

and

17

, respectively, and their centers are separated by a distance of

5

. Let

X

be a fixed intersection point of these two circles, and let

Z

be some point on

\Gamma

for which segment

XZ

intersects

\Omega

at a second point

Y

. (If

XZ

is tangent to

\Omega

, then define

Y

to be

X

.) Find the maximum possible length of segment

Y Z

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

Problem Statement