MathDB

p1. (a) Brian has a big job to do that will take him two hours to complete. He has six friends who can help him. They all work at the same rate, somewhat slower than Brian. All seven working together can finish the job in

45

minutes. How long will it take to do the job if Brian worked with only three of his friends?
(b) Brian could do his next job in

N

hours, working alone. This time he has an unlimited list of friends who can help him, but as he moves down the list, each friend works more slowly than those above on the list. The first friend would take

kN

(

k > 1

) hours to do the job alone, the second friend would take

k^2N

hours alone, the third friend would take

k^3N

hours alone, etc. Theoretically, if Brian could get all his infinite number of friends to help him, how long would it take to complete the job?
p2. (a) The centers of two circles of radius

1

are two opposite vertices of a square of side

1

. Find the area of the intersection of the two circles. (b) The centers of two circles of radius

1

are two consecutive vertices of a square of side

1

. Find the area of the intersection of the two circles and the square. (c) The centers of four circles of radius

1

are the vertices of a square of side

1

. Find the area of the intersection of the four circles.
p3. For any real number

x

[x]

denotes the greatest integer that does not exceed

x

. For example,

[7.3] = 7

[10/3] = 3

[5] = 5

. Given natural number

N

, denote as

f(N)

the following sum of

N

integers:

f(N) = [N/1] + [N/2] + [N/3] + ... + [N/n].

(a) Evaluate

f(7) - f(6)

. (b) Evaluate

f(35) - f(34)

. (c) Evaluate (with explanation)

f(1996) - f(1995)

.
p4. We will say that triangle

ABC

is good if it satisfies the following conditions:

AB = 7

, the other two sides are integers, and

\cos A =\frac27

. (a) Find the sides of a good isosceles triangle. (b) Find the sides of a good scalene (i.e. non-isosceles) triangle. (c) Find the sides of a good scalene triangle other than the one you found in (b) and prove that any good triangle is congruent to one of the three triangles you have found.
p5. (a) A bag contains nine balls, some of which are white, the others are black. Two balls are drawn at random from the bag, without replacement. It is found that the probability that the two balls are of the same color is the same as the probability that they are of different colors. How many of the nine balls were of one color and how many of the other color?
(b) A bag contains

N

balls, some of which are white, the others are black. Two balls are drawn at random from the bag, without replacement. It is found that the probability that the two balls are of the same color is the same as the probability that they are of different colors. It is also found that

180 < N < 220

. Find the exact value of

N

and determine how many of the

N

balls were of one color and how many of the other color.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

Problem Statement