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Problems > High School Math > 2002

## The Running Dog

"Yes, when I take my dog for a walk," said a mathematical friend, "he frequently supplies me with some interesting puzzle to solve. One day, for example, he waited, as I left the door, to see which way I should go, and when I started he raced along to the end of the road, immediately returning to me; again racing to the end of the road and again returning. He did this four times in all, at a uniform speed, and then ran at my side the remaining distance, which according to my paces measured 27 yards. I afterwards measured the distance from my door to the end of the road and found it to be 625 feet. Now, if I walk 4 miles per hour, what is the speed of my dog when racing to and fro?"

## Holey Sphere!

Suppose a circular hole was drilled through the center of a sphere. When the length of the hole was measured along its wall, it was found to be six inches long. What is the volume of the part of the sphere that remains after the material is removed from the hole? Express your answer as an exact real number number of cubic inches.

You don't need calculus to solve this problem (but if you know how to do it using calculus, go ahead) as long as you know the volume of a sphere is (4/3)π r³.

## What Is This Geometric Sequence?

The sum of the first few terms in a geometric progression is 11, the sum of their squares is 341, and sum of their cubes is 3641. Find the terms of the sequence.

## Have a Fibonacci Day!

Have a Fibonacci Day!

The Fibonacci numbers are defined as F0 = F1 = 1 and Fn+1 = Fn + Fn-1.

Calculate the following infinite sum:

``` F0      3F1      32F2          3nFn
---  +  ----  +  ----- + ... + ----- + ...
1        5        52           5n```

## The Long Walk

A few months ago, we had a running dog. This month it's a walking fly.

An elastic strip of length 1m is attached to a wall and the back of a car. A fly starts at the wall and crawls on the strip towards the car, which at the same time drives away from the wall at a speed of 1m/s.

Assuming that the strip can be stretched infinitely long, will the fly ever reach the back of the car?

## If You Square Me, I'll Square You!

Give two different numbers, each of which is the square of the other.

## Merry Christmas, Mr. Spencer!

An absentminded bank teller switches the dollars and cents when he cashed a check for Mr. Spencer, giving him dollars instead of cents, and cents instead of dollars. After buying a five cent newspaper, Mr. Spencer discovered he had left exactly twice as much as his original check. What was the amount of the check?