MathDB

p1. Function

f

is defined by

f (x) = ax+b

for some real values

a, b > 0

. If

f (f (x)) = 9x + 5

for all

x

, find

b

.
p2. At some point during a game, Will Avery has made

1/3

of his shots. When he shoots once and makes a basket, his average increases to

2/5

. Find his average (expressed as a fraction) after a second additional basket.
p3. A dealer has a deck of

1999

cards. He takes the top card off and “ducks” it, that is, places it on the bottom of the deck. He deals the second card onto the table. He ducks the third card, deals the fourth card, ducks the fifth card, deals the sixth card, and so forth, continuing until he has only one card left; he then ducks the last card with itself and deals it. Some of the cards (like the second and fourth cards) are not ducked at all before being dealt, while others are ducked multiple times. The question is: what is the average number of ducks per card?
p4. Point

P

lies outside circle

O

. Perpendicular lines

\ell

and m intersect at

P

. Line

\ell

is tangent to circle

O

at a point

6

units from

P

. Line

m

crosses circle

O

at a point

4

units from

P

. Find the radius of circle

O

.
p5. Define

f(n)

f(n) = \begin{cases} n/2 \,\,\,\text{if} \,\,\, n\,\,\,is\,\,\, even \\ (n + 1023)/2\,\,\, \text{if} \,\,\, n\,\,\,is\,\,\, odd \end{cases}

Find the least positive integer

n

such that

f(f(f(f(f(n))))) = n.

p6. Write

\sqrt{10001}

to the sixth decimal place, rounding down.
p7. Define

(a_n)

recursively by

a_1 = 1

a_n = 20 \cos (a_{n-1}^o)

. As

n

tends to infinity,

(a_n)

tends to

18.9195...

. Define

(b_n)

recursively by

b_1 = 1

b_n =\sqrt{800 + 800 \cos (b_{n-1}^o)}

. As

n

tends to infinity,

(b_n)

tends to

x

. Calculate

x

to three decimal places.
p8. Let

mod_d (k)

be the remainder of

k

when divided by

d

. Find the number of positive integers

n

satisfying

mod_n(1999) = n^2 - 89n + 1999

p9. Let

f(x) = x^3 + x

. Compute

\sum^{10}_{k=1} \frac{1}{1 + f^{-1}(k - 1)^2 + f^{-1}(k - 1)f^{-1}(k) + f^{-1}(k)^2}

(

f^{-1}

is the inverse of

f

f (f^{-1}1 (x)) = f^{-1}1 (f (x)) = x

for all

x

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

Problem Statement