MathDB

p1. Evaluate

\sqrt{i}

in the form

a + bi

with

a > 0

, where

a

and

b

are real numbers and

i

\sqrt{-1}

.
p2. Let

A

be the matrix

\begin{bmatrix} 3 & 2 & 1 \\ 2 & 0 & 4 \\ 0 & 6 & 3 \end{bmatrix}

.What is

det(A^{-1})

?
p3. How many positive integers divide the number of positive integers that divide

2002^{2002}

?
p4. In base

6

, how many

(2n + 1)

-digit numbers are palindromes?
p5. In the diagram below, what is the sum of five angles

\theta_i

numbered,

\sum_{n=i}^5 \theta_n

? https://cdn.artofproblemsolving.com/attachments/5/2/21d2302852680db9d2c8b31d3f01c9d0b67b56.png
p6. Let

x

be the smallest number such that

x

written out in English (i.c.

1,647

is one thousand six hundred forty seven) has exactly

300

letters. What is the most common digit

(0-9)

x

?
p7. Define

g(x) = \int_{x}^{x+1} 2^t dt

, and

g'(x) =\frac{d}{dx}g(x)

. Compute

g'(10)

.
p8. If

xy = 24

with

x

and

y

real, what is the minimum value that

x^2 + 4y^2

can attain?
p9. Find the cubic polynomial

f(x)

such that

f(1) = 1

f(2) = 5

f(3) = 14

, and

f(4) = 30

.
p10. What is

1^2 + 2^2 + 3^2 + ... + 2002^2

?
p11. The

r

-th power mean

P_r

n

numbers

x_1,...,x_n

is defined as

P_r(x_1,...,x_n) = \left(\frac{x_1^r+...+x_n^r}{n} \right)^{1/r}

for

r\ne 0

, and

P_0 = (x_1x_2 ... x_n)^{1/n}

The Power Mean Inequality says that if

r > s

, then

P_r \ge P_s

. Using this fact, find out how many ordered pairs of positive integers

(x, y)

satisfy

48\sqrt{xy}-x^2 - y^2 \ge 289

.
p12. After meeting him in the afterlife, Gauss challenges Fermat to a boxing match. Each mathematician is wearing glasses, and Gauss has a

1/3

probability of knocking off Fermat’s glasses during the match, whereas Fermat has a

1/5

chance of knocking off Gauss’s glasses. Each mathematician has a

1/2

chance of losing without his glasses and a

1/5

chance of losing anyway with his. Note that it is possible for both Fermat and Gauss to lose (simultaneous knockout) or for neither to lose (the match is a draw). Given that Gauss wins the match (and Fermat loses), what is the probability that Gauss has lost his glasses?
p13. Evaluate

\frac{1}{-1+\frac{1}{-1+\frac{1}{-1+...}}}

p14. What is the smallest positive integer

x

such that

x^2 + x + 41

is not prime?
p15. Let

A(t)

be an

n \times n

square matrix whose entries are all functions of

t

, and suppose that

det A(t)\ne 0

for all

t

. Then

\frac{dA}{dt}= A'

is simply the matrix formed by differentiating each entry of

A(t)

with respect to

t

. Write

\frac{d}{dt}(A{-1} (t))

in terms of

A(t)

and

A'

, where the only differentiation occurs in

A'

itself.
PS. You had better use hide for answers .

Problem Statement