MathDB

Round 6
p16. Given that

x

and

y

are positive real numbers such that

x^3+y = 20

, the maximum possible value of

x + y

can be written as

\frac{a\sqrt{b}}{c}

+d where

a

b

c

, and

d

are positive integers such that

gcd(a,c) = 1

and

b

is square-free. Find

a +b +c +d

.
p17. In

\vartriangle DRK

DR = 13

DK = 14

, and

RK = 15

. Let

E

be the intersection of the altitudes of

\vartriangle DRK

. Find the value of

\lfloor DE +RE +KE \rfloor

.
p18. 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 the expected number of jumps it takes Subaru to reach pad

7

.
Round 7
This set has problems whose answers depend on one another.
p19. Let

B

be the answer to Problem

20

and let

C

be the answer to Problem

21

. Given that

f (x) = x^3-Bx-C = (x-r )(x-s)(x-t )

where

r

s

, and

t

are complex numbers, find the value of

r^2+s^2+t^2

.
p20. Let

A

be the answer to Problem

19

and let

C

be the answer to Problem

21

. Circles

\omega_1

and

\omega_2

meet at points

X

and

Y

. Let point

P \ne Y

be the point on

\omega_1

such that

PY

is tangent to

\omega_2

, and let point

Q \ne Y

be the point on

\omega_2

such that

QY

is tangent to

\omega_1

. Given that

PX = A

and

QX =C

, find

XY

.
p21. Let

A

be the answer to Problem

19

and let

B

be the answer to Problem

20

. Given that the positive difference between the number of positive integer factors of

A^B

and the number of positive integer factors of

B^A

D

, and given that the answer to this problem is an odd prime, find

\frac{D}{B}-40

.
Round 8
p22. Let

v_p (n)

for a prime

p

and positive integer

n

output the greatest nonnegative integer

x

such that

p^x

divides

n

. Find

\sum^{50}_{i=1}\sum^{i}_{p=1} { v_p (i )+1 \choose 2},

where the inner summation only sums over primes

p

between

1

and

i

.
p23. Let

a

b

, and

c

be positive real solutions to the following equations.

\frac{2b^2 +2c^2 -a^2}{4}= 25

\frac{2c^2 +2a^2 -b^2}{4}= 49

\frac{2a^2 +2b^2 -c^2}{4}= 64

The area of a triangle with side lengths

a

b

, and

c

can be written as

\frac{x\sqrt{y}}{z}

where

x

and

z

are relatively prime positive integers and

y

is square-free. Find

x + y +z

.
p24. Alan, Jiji, Ina, Ryan, and Gavin want to meet up. However, none of them know when to go, so they each pick a random

1

hour period from

5

AM to

11

AM to meet up at Alan’s house. Find the probability that there exists a time when all of them are at the house at one time.
Round 9
p25. Let

n

be the number of registered participantsin this

LMT

. Estimate the number of digits of

\left[ {n \choose 2} \right]

in base

10

. If your answer is

A

and the correct answer is

C

, then your score will be

\left \lfloor \max \left( 0,20 - \left| \ln \left( \frac{A}{C}\right) \cdot 5 \right|\right| \right \rfloor.

p26. Let

\gamma

be theminimum value of

x^x

over all real numbers

x

. Estimate

\lfloor 10000\gamma \rfloor

. If your answer is

A

and the correct answer is

C

, then your score will be

\left \lfloor \max \left( 0,20 - \left| \ln \left( \frac{A}{C}\right) \cdot 5 \right|\right| \right \rfloor.

p27. Let

E = \log_{13} 1+log_{13}2+log_{13}3+...+log_{13}513513.

Estimate

\lfloor E \rfloor

. If your answer is

A

and the correct answer is

C

, your score will be

\left \lfloor \max \left( 0,20 - \left| \ln \left( \frac{A}{C}\right) \cdot 5 \right|\right| \right \rfloor.

PS. You should use hide for answers. Rounds 1-5 have been posted [url=https://artofproblemsolving.com/community/c3h3167127p28823220]here. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

Problem Statement