MathDB

Round 1
p1. A box contains

1

ball labelledW,

1

ball labelled

E

1

ball labelled

L

1

ball labelled

C

1

ball labelled

O

8

balls labelled

M

, and

1

last ball labelled

E

. One ball is randomly drawn from the box. The probability that the ball is labelled

E

\frac{1}{a}

. Find

a

.
p2. Let

G +E +N = 7

G +E +O = 15

N +T = 22.

Find the value of

T +O

.
p3. The area of

\vartriangle LMT

22

. Given that

MT = 4

and that there is a right angle at

M

, find the length of

LM

.
Round 2
p4. Kevin chooses a positive

2

-digit integer, then adds

6

times its unit digit and subtracts

3

times its tens digit from itself. Find the greatest common factor of all possible resulting numbers.
p5. Find the maximum possible number of times circle

D

can intersect pentagon

GRASS'

over all possible choices of points

G

R

A

S

, and

S'

.
p6. Find the sum of the digits of the integer solution to

(\log_2 x) \cdot (\log_4 \sqrt{x}) = 36

.
Round 3
p7. Given that

x

and

y

are positive real numbers such that

x^2 + y = 20

, the maximum possible value of

x + y

can be written as

\frac{a}{b}

where

a

and

b

are relatively prime positive integers. Find

a +b

.
p8. In

\vartriangle DRK

DR = 13

DK = 14

, and

RK = 15

. Let

E

be the point such that

ED = ER = EK

. Find the value of

\lfloor DE +RE +KE \rfloor

.
p9. Subaru the frog lives on lily pad

1

. There is a line of lily pads, numbered

2

3

4

5

6

, and

7

. Every minute, Subaru jumps from his current lily pad to a lily pad whose number is either

1

2

greater, chosen at random from valid possibilities. There are alligators on lily pads

2

and

5

. If Subaru lands on an alligator, he dies and time rewinds back to when he was on lily pad number

1

. Find how many times Subaru is expected to die before he reaches pad

7

.
Round 4
p10. Find the sum of the following series:

\sum^{\infty}_{i=1} = \frac{\sum^i_{j=1} j}{2^i}=\frac{1}{2^1}+\frac{1+2}{2^2}+\frac{1+2+3}{2^3}+\frac{1+2+3+4}{2^4}+...

p11. Let

\phi (x)

be the number of positive integers less than or equal to

x

that are relatively prime to

x

. Find the sum of all

x

such that

\phi (\phi(x)) = x -3

. Note that

1

is relatively prime to every positive integer.
p12. On a piece of paper, Kevin draws a circle. Then, he draws two perpendicular lines. Finally, he draws two perpendicular rays originating from the same point (an

L

shape). What is the maximum number of sections into which the lines and rays can split the circle?
Round 5
p13. In quadrilateral

ABCD

\angle A = 90^o

\angle C = 60^o

\angle ABD = 25^o

, and

\angle BDC = 5^o

. Given that

AB = 4\sqrt3

, the area of quadrilateral

ABCD

can be written as

a\sqrt{b}

. Find

10a +b

.
p14. The value of

\sum^6_{n=2} \left( \frac{n^4 +1}{n^4 -1}\right) -2 \sum^6_{n=2}\left(\frac{n^3 -n^2+n}{n^4 -1}\right)

can be written as

\frac{m}{n}

where

m

and

n

are relatively prime positive integers. Find

100m+n

.
p15. Positive real numbers

x

and

y

satisfy the following

2

equations.

x^{1+x^{1+x^{1+...}}}= 8

\sqrt[24]{y +\sqrt[24]{y + \sqrt[24]{y +...}}} = x

Find the value of

\lfloor y \rfloor

.
PS. You should use hide for answers. Rounds 6-9 have been posted [url=https://artofproblemsolving.com/community/c3h3167130p28823260]here. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

Problem Statement