MathDB

p1. Find the smallest positive integer

N

such that

2N -1

and

2N +1

are both composite.
p2. Compute the number of ordered pairs of integers

(a, b)

with

1 \le a, b \le 5

such that

ab - a - b

is prime.
p3. Given a semicircle

\Omega

with diameter

AB

, point

C

is chosen on

\Omega

such that

\angle CAB = 60^o

. Point

D

lies on ray

BA

such that

DC

is tangent to

\Omega

. Find

\left(\frac{BD}{BC} \right)^2

.
p4. 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)

?
p5. Regular hexagon

ABCDEF

has side length

3

. Circle

\omega

is drawn with

AC

as its diameter.

BC

is extended to intersect

\omega

at point

G

. If the area of triangle

BEG

can be expressed as

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

for positive integers

a, b, c

with

b

squarefree and

gcd(a, c) = 1

, find

a + b + c

.
p6. Suppose that

x

and

y

are positive real numbers such that

\log_2 x = \log_x y = \log_y 256

. Find

xy

.
p7. 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.
p8. Let

\vartriangle ABC

be an equilateral triangle. Isosceles triangles

\vartriangle DBC

\vartriangle ECA

, and

\vartriangle FAB

, not overlapping

\vartriangle ABC

, are constructed such that each has area seven times the area of

\vartriangle ABC

. Compute the ratio of the area of

\vartriangle DEF

to the area of

\vartriangle ABC

.
p9. Consider the sequence of polynomials an(x) with

a_0(x) = 0

a_1(x) = 1

, and

a_n(x) = a_{n-1}(x) + xa_{n-2}(x)

for all

n \ge 2

. Suppose that

p_k = a_k(-1) \cdot a_k(1)

for all nonnegative integers

k

. Find the number of positive integers

k

between

10

and

50

, inclusive, such that

p_{k-2} + p_{k-1} = p_{k+1} - p_{k+2}

.
p10. In triangle

ABC

, point

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

.
p11. 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.
p12. 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.

Problem Statement