Problem of the Fortnight

Problem of the Fortnight

 

The problem of the week is back, but in a FORTNIGHT version! Similarly as in the previous semesters, a small prize will be awarded to the random correct solver, and a grand prize will be awarded at the end of the semester to the solver with the most correct submissions (or a random winner will be drawn in case of a tie).

Solutions (with explanations!) are to be submitted to Prof. Pavel Bělík, either electronically to belik@augsburg.edu, or to his mailbox or office in SCI 137A.

 Problem of the Fortnight #3:

 

A rectangle of unknown dimensions is cut into four pieces as shown in the figure below (not necessarily up to scale). Two of the four pieces have areas of 2 and 4 units as shown. Can the areas of the two other pieces be determined? If so, what are they?

POTF03-300x205

They are due by Friday, October 25, 2013 (midterm break).

 

 

Problem of the Fortnight #2:

 

Four cities, Angerville, Blabberville, Clutterville, and Deville, are located in the corners of a rectangle with sides 8 miles and 6 miles as shown in the left figure below. You are supposed to build a road (or roads) that allows you to travel from any one city to any other city. How long is the shortest road you can construct?

(An example might be a circle of radius 5 as shown in the second figure. The total length of this road is 2πr = 10π miles. Can you do better than that? Shortest solution wins!)

POTF02-300x181

Submit your solution (with explanations!)  by Friday, October 11, 2013.

Problem of the Fortnight #1:

 

In a survey people are asked: “Do you think the new president will be better than the last one?”

Suppose a people say “better”, b say “the same” and c say “worse.” Sociologists calculate two measures of “social optimism”: SO­1 = a + b/2 and SO2 = a - c. If 100 people respond to the survey and SO1 = 40, find SO2.

Submit your solution (with explanations!) by Friday, September 27, 2013.