MathDB

p1. Given an equilateral triangle

ABC

with area

16\sqrt3

, and an interior point

P

with distances from vertices

|AP| = 4

and

|BP| = 6

. (a) Find the length of each side. (b) Find the distance from point

P

to the side

AB

. (c) Find the distance

|PC|

.
p2. Several players play the following game. They form a circle and each in turn tosses a fair coin. If the coin comes up heads, that player drops out of the game and the circle becomes smaller, if it comes up tails that player remains in the game until his or her next turn to toss. When only one player is left, he or she is the winner. For convenience let us name them

A

(who tosses first),

B

(second),

C

(third, if there is a third), etc. (a) If there are only two players, what is the probability that

A

(the first) wins? (b) If there are exactly

3

players, what is the probability that

A

(the first) wins? (c) If there are exactly

3

players, what is the probability that

B

(the second) wins?
p3. A circular castle of radius

r

is surrounded by a circular moat of width

m

(

m

is the shortest distance from each point of the castle wall to its nearest point on shore outside the moat). Life guards are to be placed around the outer edge of the moat, so that at least one life guard can see anyone swimming in the moat. (a) If the radius

r

140

feet and there are only

3

life guards available, what is the minimum possible width of moat they can watch? (b) Find the minimum number of life guards needed as a function of

r

and

m

. https://cdn.artofproblemsolving.com/attachments/a/8/d7ff0e1227f9dcf7e49fe770f7dae928581943.png
p4. (a)Find all linear (first degree or less) polynomials

f(x)

with the property that

f(g(x)) = g(f(x))

for all linear polynomials

g(x)

. (b) Prove your answer to part (a). (c) Find all polynomials

f(x)

with the property that

f(g(x)) = g(f(x))

for all polynomials

g(x)

. (d) Prove your answer to part (c).
p5. A non-empty set

B

of integers has the following two properties: i. each number

x

in the set can be written as a sum

x = y+ z

for some

y

and

z

in the set

B

. (Warning:

y

and

z

may or may not be distinct for a given

x

.) ii. the number

0

can not be written as a sum

0 = y + z

for any

y

and

z

in the set

B

.
(a) Find such a set

B

with exactly

6

elements. (b) Find such a set

B

with exactly

6

elements, and such that the sum of all the

6

elements is

1988

. (c) What is the smallest possible size of such a set

B

? (d) Prove your answer to part (c).
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

Problem Statement