MathDB

Round 9
p25. Let

a

b

, and

c

be positive numbers with

a +b +c = 4

. If

a,b,c \le 2

and

M =\frac{a^3 +5a}{4a^2 +2}+\frac{b^3 +5b}{4b^2 +2}+\frac{c^3 +5c}{4c^2 +2},

then find the maximum possible value of

\lfloor 100M \rfloor

.
p26. In

\vartriangle ABC

AB = 15

AC = 16

, and

BC = 17

. Points

E

and

F

are chosen on sides

AC

and

AB

, respectively, such that

CE = 1

and

BF = 3

. A point

D

is chosen on side

BC

, and let the circumcircles of

\vartriangle BFD

and

\vartriangle CED

intersect at point

P \ne D

. Given that

\angle PEF = 30^o

, the length of segment

PF

can be expressed as

\frac{m}{n}

. Find

m+n

.
p27. Arnold and Barnold are playing a game with a pile of sticks with Arnold starting first. Each turn, a player can either remove

7

sticks or

13

sticks. If there are fewer than

7

sticks at the start of a player’s turn, then they lose. Both players play optimally. Find the largest number of sticks under

200

where Barnold has a winning strategy
Round 10
p28. Let

a

b

, and

c

be positive real numbers such that

\log_2(a)-2 = \log_3(b) =\log_5(c)

and

a +b = c

. What is

a +b +c

?
p29. Two points,

P(x, y)

and

Q(-x, y)

are selected on parabola

y = x^2

such that

x > 0

and the triangle formed by points

P

Q

, and the origin has equal area and perimeter. Find

y

.
p30.

5

families are attending a wedding.

2

families consist of

4

people,

2

families consist of

3

people, and

1

family consists of

2

people. A very long row of

25

chairs is set up for the families to sit in. Given that all members of the same family sit next to each other, let the number of ways all the people can sit in the chairs such that no two members of different families sit next to each other be

n

. Find the number of factors of

n

.
Round 11
p31. Let polynomial

P(x) = x^3 +ax^2 +bx +c

have (not neccessarily real) roots

r_1

r_2

, and

r_3

. If

2ab = a^3 -20 = 6c -21

, then the value of

|r^3_1+r^3_2+r^3_3|

can be written as

\frac{m}{n}

where

m

and

n

are relatively prime positive integers. Find the value of

m+n

.
p32. In acute

\vartriangle ABC

, let

H

I

O

, and

G

be the orthocenter, incenter, circumcenter, and centroid of

\vartriangle ABC

, respectively. Suppose that there exists a circle

\omega

passing through

B

I

H

, and

C

, the circumradius of

\vartriangle ABC

312

, and

OG = 80

. Let

H'

, distinct from

H

, be the point on

\omega

such that

\overline{HH'}

is a diameter of

\omega

. Given that lines

H'O

and

BC

meet at a point

P

, find the length

OP

.

p33. Find the number of ordered quadruples

(x, y, z,w)

such that

0 \le x, y, z,w \le 1000

are integers and

x!+ y! =2^z \cdot w!

holds (Note:

0! = 1

).
Round 12
p34. Let

Z

be the product of all the answers from the teams for this question. Estimate the number of digits of

Z

. If your estimate is

E

and the answer is

A

, your score for this problem will be

\max \left( 0, \lceil 15- |A-E| \rceil \right).

Your answer must be a positive integer.
p35. Let

N

be number of ordered pairs of positive integers

(x, y)

such that

3x^2 -y^2 = 2

and

x < 2^{75}

. Estimate

N

. If your estimate is

E

and the answer is

A

, your score for this problem will be

\max \left( 0, \lceil 15- 2|A-E| \rceil \right).

p36.

30

points are located on a circle. How many ways are there to draw any number of line segments between the points such that none of the line segments overlap and none of the points are on more than one line segment? (It is possible to draw no line segments). If your estimate is

E

and the answer is

A

, your score for this problem will be

\max \left( 0, \left \lceil 15- \ln \frac{A}{E} \right \rceil \right).

PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h3166472p28814057]here and 5-8 [url=https://artofproblemsolving.com/community/c3h3166476p28814111]here.. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

Problem Statement