MathDB

Round 1
p1. Solve the maze https://cdn.artofproblemsolving.com/attachments/8/c/6439816a52b5f32c3cb415e2058556edb77c80.png
p2. Billiam can write a problem in

30

minutes, Jerry can write a problem in

10

minutes, and Evin can write a problem in

20

minutes. Billiam begins writing problems alone at

3:00

PM until Jerry joins himat

4:00

PM, and Evin joins both of them at

4:30

PM. Given that they write problems until the end of math team at

5:00

PM, how many full problems have they written in total?
p3. How many pairs of positive integers

(n,k)

are there such that

{n \choose k}= 6

?
Round 2
p4. Find the sumof all integers

b > 1

such that the expression of

143

in base

b

has an even number of digits and all digits are the same.
p5. Ιni thinks that

a \# b = a^2 - b

and

a \& b = b^2 - a

, while Mimi thinks that

a \# b = b^2 - a

and

a \& b = a^2 - b

. Both Ini and Mimi try to evaluate

6 \& (3 \# 4)

, each using what they think the operations

\&

and

\#

mean. What is the positive difference between their answers?
p6. A unit square sheet of paper lies on an infinite grid of unit squares. What is the maximum number of grid squares that the sheet of paper can partially cover at once? A grid square is partially covered if the area of the grid square under the sheet of paper is nonzero - i.e., lying on the edge only does not count.
Round 3
p7. Maya wants to buy lots of burgers. A burger without toppings costs

\$4

, and every added topping increases the price by 50 cents. There are 5 different toppings for Maya to choose from, and she can put any combination of toppings on each burger. How much would it cost forMaya to buy

1

burger for each distinct set of toppings? Assume that the order in which the toppings are stacked onto the burger does not matter.
p8. Consider square

ABCD

and right triangle

PQR

in the plane. Given that both shapes have area

1

PQ =QR

PA = RB

, and

P

A

B

and

R

are collinear, find the area of the region inside both square

ABCD

and

\vartriangle PQR

, given that it is not

0

.
p9. Find the sum of all

n

such that

n

is a

3

-digit perfect square that has the same tens digit as

\sqrt{n}

, but that has a different ones digit than

\sqrt{n}

.
Round 4
p10. Jeremy writes the string:

LMTLMTLMTLMTLMTLMT

on a whiteboard (“

LMT

” written

6

times). Find the number of ways to underline

3

letters such that from left to right the underlined letters spell LMT.
p11. Compute the remainder when

12^{2022}

is divided by

1331

.
p12. What is the greatest integer that cannot be expressed as the sum of

5

23

s, and

29

s?
Round 5
p13. Square

ABCD

has point

E

on side

BC

, and point

F

on side

CD

, such that

\angle EAF = 45^o

. Let

BE = 3

, and

DF = 4

. Find the length of

FE

.
p14. Find the sum of all positive integers

k

such that

\bullet

k

is the power of some prime.

\bullet

k

can be written as

5654_b

for some

b > 6

.
p15. If

\sqrt[3]{x} + \sqrt[3]{y} = 2

and

x + y = 20

, compute

\max \,(x, y)

.
PS. You should use hide for answers. Rounds 6-9 have been posted [url=https://artofproblemsolving.com/community/c3h3167372p28825861]here. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

Problem Statement