MathDB

p1. Suppose that

x

and

y

are positive real numbers such that

\log_2 x = \log_x y = \log_y 256

. Find

xy

.
p2. Let the roots of

x^2 + 7x + 11

r

and

s

. If f(x) is the monic polynomial with roots

rs + r + s

and

r^2 + s^2

, what is

f(3)

?
p3. Call a positive three digit integer

\overline{ABC}

fancy if

\overline{ABC} = (\overline{AB})^2 - 11 \cdot \overline{C}

. Find the sum of all fancy integers.
p4. In triangle

ABC

, points

D

and

E

are on line segments

BC

and

AC

, respectively, such that

AD

and

BE

intersect at

H

. Suppose that

AC = 12

BC = 30

, and

EC = 6

. Triangle

BEC

has area

45

and triangle

ADC

has area

72

, and lines

CH

and

AB

meet at

F

. If

BF^2

can be expressed as

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

for positive integers

a

b

c

d

with

c

squarefree and

gcd(a, b, d) = 1

, then find

a + b + c + d

.
p5. Find the minimum possible integer

y

such that

y > 100

and there exists a positive integer

x

such that

x^2 + 18x + y

is a perfect fourth power.
p6. Let

ABCD

be a quadrilateral such that

AB = 2

CD = 4

BC = AD

, and

\angle ADC + \angle BCD = 120^o

. If the sum of the maximum and minimum possible areas of quadrilateral

ABCD

can be expressed as

a\sqrt{b}

for positive integers

a, b

with

b

squarefree, then find

a + b

.
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

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