## Analysis

An archive of questions and answers that may be of interest to puzzle enthusiasts.
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Question 1 - bicycle:
A boy, a girl and a dog go for a 10 mile walk. The boy and girl can walk at 2 mph and the dog can trot at 4 mph. They also have a bicycle which only one of them (including the dog!) can use at a time. When riding, the boy and girl can travel at 12 mph while the dog can pedal at 16 mph. What is the shortest time in which all three can complete the trip? Show Answer

Question 2 - boy.girl.dog:
A boy, a girl and a dog are standing together on a long, straight road. Simulataneously, they all start walking in the same direction: The boy at 4 mph, the girl at 3 mph, and the dog trots back and forth between them at 10 mph. Assume all reversals of direction instantaneous. In one hour, where is the dog and in which direction is he facing? Show Answer

Question 3 - bugs:
Four bugs are placed at the corners of a square. Each bug walks always directly toward the next bug in the clockwise direction. How far do the bugs walk before they meet? Show Answer

Question 4 - infinity:
What function is zero at zero, strictly positive elsewhere, infinitely differentiable at zero and has all zero derivatives at zero? Show Answer

Question 5 - cache:
Cache and Ferry (How far can a truck go in a desert?) A pick-up truck is in the desert beside N 50-gallon gas drums, all full. The truck's gas tank holds 10 gallons and is empty. The truck can carry one drum, whether full or empty, in its bed. It gets 10 miles to the gallon. How far away from the starting point can you drive the truck? Show Answer

Question 6 - calculate.pi
How can I calculate many digits of pi? Show Answer

Question 7: cats.and.rats
If 6 cats can kill 6 rats in 6 minutes, how many cats does it take to kill one rat in one minute?

Question 8: dog
A body of soldiers form a 50m-by-50m square ABCD on the parade ground. In a unit of time, they march forward 50m in formation to take up the position DCEF. The army's mascot, a small dog, is standing next to its handler at location A. When the soldiers start marching, the dog begins to run around the moving body in a clockwise direction, keeping as close to it as possible. When one unit of time has elapsed, the dog has made one complete circuit and has got back to its handler, who is now at location D. (We can assume the dog runs at a constant speed and does not delay when turning the corners.)

```         B----C----E
|    |    | forward-->
A----D----F
```
How far does the dog travel? Show Answer

Question 9: e.and.pi
Without finding their numerical values, which is greater, e^(pi) or (pi)^e? Show Answer

Question 10: functional/distributed
Find all f: R -> R, f not identically zero, such that
(*) f( (x+y)/(x-y) ) = ( f(x)+f(y) )/( f(x)-f(y) ). Show Answer

Question 11: functional/linear
Suppose f is non-decreasing with
f(x+y) = f(x) + f(y) + C for all real x, y.

Prove: there is a constant A such that f(x) = Ax - C for all x.

Question 12 - integral:
If f is integrable on (0,inf) and differentiable at 0, and a > 0, and:

inf
Int (f(x)/x) dx is defined
0

show:

inf
Int (( f(x) - f(ax) )/x) dx = f(0) ln(a)

Question 14 - irrational.stamp:
You have an ink stamp which is so amazingly precise that, when inked and pressed down on the plane, it makes every circle of irrational radius (centered at the center of the stamp) black.

Question: Can one use the stamp three times and make every point in the plane black? [assume plane was white to begin with, and ignore the fact that no such stamp is physically possible]

Question 15 - minimum.time:
N people can walk or drive in a two-seater to go from city A to city B. What is the minimum time required to do so? Show Answer

Question 16 - particle:
What is the longest time that a particle can take in travelling between two points if it never increases its acceleration along the way and reaches the second point with speed V? Show Answer

Question 17 - period:
What is the least possible integral period of the sum of functions of periods 3 and 6? Show Answer

Question 18 - rubberband
A bug walks down a rubber band which is attached to a wall at one end and a car moving away from the wall at the other end. The car is moving at 1 m/sec while the bug is only moving at 1 cm/sec. Assuming the rubber band is uniformly and infinitely elastic, will the bug ever reach the car? Show Answer

Question 19 - sequence:
Show that in the sequence: x, 2x, 3x, .... (n-1)x (x can be any real number) there is at least one number which is within 1/n of an integer. Show Answer

Question 20 - snow:
Snow starts falling before noon on a cold December day. At noon a snow plow starts plowing a street. It travels 1 mile in the first hour, and 1/2 mile in the second hour. What time did the snow start falling??

You may assume that the plow's rate of travel is inversely proportional to the height of the snow, and that the snow falls at a uniform rate. Show Answer

Question 21 - tower:
R = N ^ (N ^ (N ^ ...)). What is the maximum N>0 that will yield a finite R? Show Answer