1, 101, 10101, 1010101, ...

Show that, in the sequence 1, 101, 10101, 1010101, ... only the number 101 is prime.

Arctangent sum

Evaluate the sum of arctan(1/(2n^2)) from n=1 to infinity.

Line joining tangent circles

A circle of radius 4 is tangent to a circle of radius 9, and a line is tangent to both circles at A and B. Find |AB|.

x^y = y^x

If 0 < x < y and x^y = y^x then what is the greatest possible value (least upper bound) of x?

3D Tic-Tac-Toe

In how many ways can you get three in a row in three-dimensional tic-tac-toe (noughts and crosses)?

Colorado balloon incident

I gave my students an assignment related to the Colorado balloon incident. Any comments or suggestions would be appreciated. Are there other questions that should have been included? The last question seems to be unanswerable with the given information, but I am including it anyway.

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Read the excerpt from the Wikipedia article on the Colorado Balloon Incident, and then answer the following questions.

1. What is the volume of the balloon, in cubic meters?
2. How many cubic feet are in one cubic meter?
3. What is the lift capacity of helium at sea level at 0° C?
4. The article states that the balloon would be sufficient to lift 29 kg at sea level. Show how this would be calculated.
5. How high above Fort Collins did the balloon reach?
6. How much helium would have been needed to carry Falcon Heene to an altitude of 8,000 feet?

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Note: These questions are released to the public domain, and may be used or modified without attribution.

f(f(x)) = x^2

Find a real function f so that f(f(x)) = x^2 for all x.