MathDB

p1. Let

A\% B = BA - B - A + 1

. How many digits are in the number

1\%(3\%(3\%7))

?
p2. Three circles, of radii

1, 2

, and

3

are all externally tangent to each other. A fourth circle is drawn which passes through the centers of those three circles. What is the radius of this larger circle?
p3. Express

\frac13

in base

2

as a binary number. (Which, similar to how demical numbers have a decimal point, has a “binary point”.)
p4. Isosceles trapezoid

ABCD

with

AB

parallel to

CD

is constructed such that

DB = DC

. If

AD = 20

AB = 14

, and

P

is the point on

AD

such that

BP + CP

is minimized, what is

AP/DP

?
p5. Let

f(x) = \frac{5x-6}{x-2}

. Define an infinite sequence of numbers

a_0, a_1, a_2,....

such that

a_{i+1} = f(a_i)

and

a_i

is always an integer. What are all the possible values for

a_{2014}

?
p6.

MATH

and

TEAM

are two parallelograms. If the lengths of

MH

and

AE

are

13

and

15

, and distance from

AM

T

12

, find the perimeter of

AMHE

.
p7. How many integers less than

1000

are there such that

n^n + n

is divisible by

5

?
p8.

10

coins with probabilities of

1, 1/2, 1/3 ,..., 1/10

of coming up heads are flipped. What is the probability that an odd number of them come up heads?
p9. An infinite number of coins with probabilities of

1/4, 1/9, 1/16, ...

of coming up heads are all flipped. What is the probability that exactly

1

of them comes up heads?
p10. Quadrilateral

ABCD

has side lengths

AB = 10

BC = 11

, and

CD = 13

. Circles

O_1

and

O_2

are inscribed in triangles

ABD

and

BDC

. If they are both tangent to

BD

at the same point

E

, what is the length of

DA

?
PS. You had better use hide for answers.

Problem Statement