MathDB

p1. Let

a

and

b

be complex numbers such that

a^3 + b^3 = -17

and

a + b = 1

. What is the value of

ab

?
p2. Let

AEFB

be a right trapezoid, with

\angle AEF = \angle EAB = 90^o

. The two diagonals

EB

and

AF

intersect at point

D

, and

C

is a point on

AE

such that

AE \perp DC

. If

AB = 8

and

EF = 17

, what is the lenght of

CD

?
p3. How many three-digit numbers

abc

(where each of

a

b

, and

c

represents a single digit,

a \ne 0

) are there such that the six-digit number

abcabc

is divisible by

2

3

5

7

11

, or

13

?
p4. Let

S

be the sum of all numbers of the form

\frac{1}{n}

where

n

is a postive integer and

\frac{1}{n}

terminales in base

b

, a positive integer. If

S

\frac{15}{8}

, what is the smallest such

b

?
p5. Sysyphus is having an birthday party and he has a square cake that is to be cut into

25

square pieces. Zeus gets to make the first straight cut and messes up badly. What is the largest number of pieces Zeus can ruin (cut across)? Diagram?
p6. Given

(9x^2 - y^2)(9x^2 + 6xy + y^2) = 16

and

3x + y = 2

. Find

x^y

.
p7. What is the prime factorization of the smallest integer

N

such that

\frac{N}{2}

is a perfect square,

\frac{N}{3}

is a perfect cube,

\frac{N}{5}

is a perfect fifth power?
p8. What is the maximum number of pieces that an spherical watermelon can be divided into with four straight planar cuts?
p9. How many ordered triples of integers

(x,y,z)

are there such that

0 \le x, y, z \le 100

and

(x - y)^2 + (y - z)^2 + (z - x)^2 \ge (x + y - 2z) + (y + z - 2x)^2 + (z + x - 2y)^2.

p10. Find all real solutions to

(2x - 4)^2 + (4x - 2)^3 = (4x + 2x - 6)^3

.
p11. Let

f

be a function that takes integers to integers that also has

f(x)=\begin{cases} x - 5\,\, if \,\, x \ge 50 \\ f (f (x + 12)) \,\, if \,\, x < 50 \end{cases}

Evaluate

f (2) + f (39) + f (58).

p12. If two real numbers are chosen at random (i.e. uniform distribution) from the interval

[0,1]

, what is the probability that theit difference will be less than

\frac35

?
p13. Let

a

b

, and

c

be positive integers, not all even, such that

a < b

b = c - 2

, and

a^2 + b^2 = c^2

. What is the smallest possible value for

c

?
p14. Let

ABCD

be a quadrilateral whose diagonals intersect at

O

. If

BO = 8

OD = 8

AO = 16

OC = 4

, and

AB = 16

, then find

AD

.
p15. Let

P_0

be a regular icosahedron with an edge length of

17

units. For each nonnegative integer

n

, recursively construct

P_{n+1}

from Pn by performing the following procedure on each face of

P_n

: glue a regular tetrahedron to that face such that three of the vertices of the tetrahedron are the midpoints of the three adjacent edges of the face, and the last vertex extends outside of

P_n

. Express the number of square units in the surface area of

P_{17}

in the form

\frac{u^v\cdot w \sqrt{x}}{y^z}

, where

u, v, w, x, y

, and

z

are integers, all greater than or equal to

2

, that satisfy the following conditions: the only perfect square that evenly divides

x

1

, the GCD of

u

and y is

1

, and neither

u

nor

y

divides

w

. Answers written in any other form will not be considered correct!
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

Problem Statement