MathDB

Round 4
p10. How many divisors of

10^{11}

have at least half as many divisors that

10^{11}

has?
p11. Let

f(x, y) = \frac{x}{y}+\frac{y}{x}

and

g(x, y) = \frac{x}{y}-\frac{y}{x}

. Then, if

\underbrace{f(f(... f(f(}_{2021 fs} f(f(1, 2), g(2,1)), 2), 2)... , 2), 2)

can be expressed in the form

a + \frac{b}{c}

, where

a

b

c

are nonnegative integers such that

b < c

and

gcd(b,c) = 1

, find

a + b + \lceil (\log_2 (\log_2 c)\rceil

p12. Let

ABC

be an equilateral triangle, and let

DEF

be an equilateral triangle such that

D

E

, and

F

lie on

AB

BC

, and

CA

, respectively. Suppose that

AD

and

BD

are positive integers, and that

\frac{[DEF]}{[ABC]}=\frac{97}{196}

. The circumcircle of triangle

DEF

meets

AB

BC

, and

CA

again at

G

H

, and

I

, respectively. Find the side length of an equilateral triangle that has the same area as the hexagon with vertices

D, E, F, G, H

, and

I

.
Round 5
p13. Point

X

is on line segment

AB

such that

AX = \frac25

and

XB = \frac52

. Circle

\Omega

has diameter

AB

and circle

\omega

has diameter

XB

. A ray perpendicular to

AB

begins at

X

and intersects

\Omega

at a point

Y

. Let

Z

be a point on

\omega

such that

\angle YZX = 90^o

. If the area of triangle

XYZ

can be expressed as

\frac{a}{b}

for positive integers

a, b

with

gcd(a, b) = 1

, find

a + b

.
p14. Andrew, Ben, and Clayton are discussing four different songs; for each song, each person either likes or dislikes that song, and each person likes at least one song and dislikes at least one song. As it turns out, Andrew and Ben don't like any of the same songs, but Clayton likes at least one song that Andrew likes and at least one song that Ben likes! How many possible ways could this have happened?
p15. Let triangle

ABC

with circumcircle

\Omega

satisfy

AB = 39

BC = 40

, and

CA = 25

. Let

P

be a point on arc

BC

not containing

A

, and let

Q

and

R

be the reflections of

P

AB

and

AC

, respectively. Let

AQ

and

AR

meet

\Omega

again at

S

and

T

, respectively. Given that the reflection of

QR

over

BC

is tangent to

\Omega

ST

can be expressed as

\frac{a}{b}

for positive integers

a, b

with

gcd(a,b)= 1

. Find

a + b

.
PS. You should use hide for answers. Rounds 1-3 have been posted [url=https://artofproblemsolving.com/community/c4h3131401p28368159]here and 6-7 [url=https://artofproblemsolving.com/community/c4h3131434p28368604]here ,Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

Problem Statement