MathDB

p1. Given that the three points where the parabola

y = bx^2 -2

intersects the

x

-axis and

y

-axis form an equilateral triangle, compute

b

.
p2. Compute the last digit of

2^{(3^{(4^{...2014)})})}

p3. A math tournament has a test which contains

10

questions, each of which come from one of three different subjects. The subject of each question is chosen uniformly at random from the three subjects, and independently of the subjects of all the other questions. The test is unfair if any one subject appears at least 5 times. Compute the probability that the test is unfair.
p4. Let

S_n

be the sum

S_n = 1 + 11 + 111 + 1111 + ... + 111... 11

where the last number

111...11

has exactly

n

1

’s. Find

\lfloor 10^{2017}/S_{2014} \rfloor

.
p5.

ABC

is an equilateral triangle with side length

12

. Let

O_A

be the point inside

ABC

that is equidistant from

B

and

C

and is

\sqrt3

units from

A

. Define

O_B

and

O_C

symmetrically. Find the area of the intersection of triangles

O_ABC

AO_BC

, and

ABO_C

.
p6. A composition of a natural number

n

is a way of writing it as a sum of natural numbers, such as

3 = 1 + 2

. Let

P(n)

denote the sum over all compositions of

n

of the number of terms in the composition. For example, the compositions of

3

are

3

1+2

2+1

, and

1+1+1

; the first has one term, the second and third have two each, and the last has

3

terms, so

P(3) = 1 + 2 + 2 + 3 = 8

. Compute

P(9)

.
p7. Let

ABC

be a triangle with

AB = 7

AC = 8

, and

BC = 9

. Let the angle bisector of

A

intersect

BC

D

. Let

E

be the foot of the perpendicular from

C

to line

AD

. Let

M

be the midpoint of

BC

. Find

ME

.
p8. Call a function

g

lower-approximating for

f

on the interval

[a, b]

if for all

x \in [a, b]

f(x) \ge g(x)

. Find the maximum possible value of

\int_1^2 g(x)dx

where

g(x)

is a linear lower-approximating function for

f(x) = x^x

[1, 2]

.
p9. Determine the smallest positive integer

x

such that

1.24x

is the same number as the number obtained by taking the first (leftmost) digit of

x

and moving it to be the last (rightmost) digit of

x

.
p10. Let

a

and

b

be real numbers chosen uniformly and independently at random from the interval

[-10, 10]

. Find the probability that the polynomial

x^5+ax+b

has exactly one real root (ignoring multiplicity).
p11. Let

b

be a positive real number, and let

a_n

be the sequence of real numbers defined by

a_1 = a_2 = a_3 = 1

, and

a_n = a_{n-1} + a_{n-2} + ba_{n-3}

for all

n > 3

. Find the smallest value of

b

such that

\sum_{n=1}^{\infty} \frac{\sqrt{a_n}}{2^n}

diverges.
p12. Find the smallest

L

such that

\left( 1 - \frac{1}{a} \right)^b \left( 1 - \frac{1}{2b} \right)^c \left( 1 - \frac{1}{3c} \right)^a \le L

for all real numbers

a, b

, and

c

greater than

1

.
p13. Find the number of distinct ways in which

30^{(30^{30})}

can be written in the form

a^{(b^c)}

, where

a, b

, and

c

are integers greater than

1

.
p14. Convex quadrilateral

ABCD

has sidelengths

AB = 7

BC = 9

CD = 15

. A circle with center

I

lies inside the quadrilateral, and is tangent to all four of its sides. Let

M

and

N

be the midpoints of

AC

and

BD

, respectively. It can be proven that

I

always lies on segment

MN

. If

I

is in fact the midpoint of

MN

, find the area of quadrilateral

ABCD

.
p15. Marc has a bag containing

10

balls, each with a different color. He draws out two balls uniformly at random and then paints the first ball he drew to match the color of the second ball. Then he places both balls back in the bag. He repeats until all the balls are the same color. Compute the expected number of times Marc has to perform this procedure before all the balls are the same color.
PS. You had better use hide for answers.

Problem Statement