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, 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?


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!)


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.