Past Problem of the Week

Problem of the Week Information:

  • Solutions to be submitted to Professor Pavel Belik (paper copies are allowed, or e-mail (belik@augsburg.edu) with solutions/explanations.
  • From the (most) correct/complete solutions, a weekly winner will be drawn and awarded a small prize.
  • The solver with the most points at the end of the semester will be awarded a Grand Prize (hopefully not in name only). :)

Posted November 26, 2012/ Solutions due November 30, 2012

(a) Show that the expression sqrt(2+ sqrt(2+ sqrt(2+ sqrt(2+ …, is equal to an integer. What is this integer?

(b) For what other integer values of  does the expression sqrt(n+ sqrt(n+ sqrt(n+ sqrt(n+ …, equal an integer?

Previous Problems

We are not accepting solutions to these problems, but they are good practice!


Posted November 16, 2012/ Solutions due November 23, 2012

A rubber band is stretched around two points 2 inches apart and a coin of diameter 1 inch centered exactly between the two points (see the figure). Exactly how long is the stretched rubber band?

potw11


Posted November 3, 2012 / Solutions due November 9, 2012

Divide the figure below into 4 congruent pieces such that each piece contains each of the letters A, B, C, and D:

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** Special Bonus Problem ** Posted November 3, 2012/ Solutions due November 9, 2012

I am a number with the following properties:

  1. If I am not a multiple of 4, then I am between 60 and 69.
  2. If I am a multiple of 3, then I am between 50 and 59.
  3. If I am not a multiple of 6, then I am between 70 and 79.

What number am I?


Posted October 22, 2012 / Solutions due October 26, 2012

 

In Calculus I you learn that the derivative of a product of two functions is not the product of their derivatives, i.e.,

(f(x)*g(x))’ is not equal to f’(x)*g’(x),

but you have to use the product rule to find the derivative. However, if either f(x)=0 or g(x)=0, then (f(x)*g(x))’=f’(x)*g’(x) is true since both sides are equal to 0. Are there any nonzero functions and for which this “wrong product rule” holds?


Posted October 12, 2012 / Solutions due October 19, 2012

How many different squares can be found in the picture below? How many different triangles?

potw6


Posted October 5, 2012 / Solutions due October 12, 2012

When 30 children in a classroom line up for lunch, Ann insists on being somewhere ahead of Betty, who insists on being somewhere ahead of Carol. If both demands are to be satisfied, in how many ways can the children line up?


Posted September 28, 2012 / Solutions due October 5, 2012

A box contains 100 socks – some are white and the rest are black. What is the number of socks of each color if the probability of randomly drawing a pair of the same color is exactly ½? What is the answer if the box contains 100,000,000 socks? What about 1,000 socks?


Posted September 21, 2012 / Solutions Due September 28, 2012

What is the largest 10-digit integer divisible by 11 containing all the digits 0-9?


Posted September 17, 2012 / Solutions Due September 21, 2012

You would like to be able to weigh any object whose possible mass is 1 ounce, or 2 ounces, …, or 40 ounces. You have at your disposal a balance scale and four weights. What should be the masses of these weights? The weights may be used on either or both sides of the scale.


Posted September 7, 2012

“Adding, subtracting, multiplying and dividing two positive integers, we get four results. Adding these four results, we get 243. What are the two integers?”