MathDB

p1. The entrance fee the county fair is

64

cents. Unfortunately, you only have nickels and quarters so you cannot give them exact change. Furthermore, the attendent insists that he is only allowed to change in increments of six cents. What is the least number of coins you will have to pay?
p2. At the county fair, there is a carnival game set up with a mouse and six cups layed out in a circle. The mouse starts at position

A

and every ten seconds the mouse has equal probability of jumping one cup clockwise or counter-clockwise. After a minute if the mouse has returned to position

A

, you win a giant chunk of cheese. What is the probability of winning the cheese?
p3. A clown stops you and poses a riddle. How many ways can you distribute

21

identical balls into

3

different boxes, with at least

4

balls in the first box and at least

1

ball in the second box?
p4. Watch out for the pig. How many sets

S

of positive integers are there such that the product of all the elements of the set is

125970

?
p5. A good word is a word consisting of two letters

A

B

such that there is never a letter

B

between any two

A

's. Find the number of good words with length

8

.
p6. Evaluate

\sqrt{2 -\sqrt{2 +\sqrt{2-...}}}

without looking.
p7. There is nothing wrong with being odd. Of the first

2006

Fibonacci numbers (

F_1 = 1

F_2 = 1

), how many of them are even?
p8. Let

f

be a function satisfying

f (x) + 2f (27- x) = x

. Find

f (11)

.
p9. Let

A

B

C

denote digits in decimal representation. Given that

A

is prime and

A -B = 4

, nd

(A,B,C)

such that

AAABBBC

is a prime.
p10. Given

\frac{x^2+y^2}{x^2-y^2} + \frac{x^2-y^2}{x^2+y^2} = k

, find

\frac{x^8+y^8}{x^8-y^8}

in term of

k

.
p11. Let

a_i \in \{-1, 0, 1\}

for each

i = 1, 2, 3, ..., 2007

. Find the least possible value for

\sum^{2006}_{i=1}\sum^{2007}_{j=i+1} a_ia_j

.
p12. Find all integer solutions

x

x^2 + 615 = 2^n

for any integer

n \ge 1

.
p13. Suppose a parabola

y = x^2 - ax - 1

intersects the coordinate axes at three points

A

B

, and

C

. The circumcircle of the triangle

ABC

intersects the

y

- axis again at point

D = (0, t)

. Find the value of

t

.
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

Problem Statement