BGSU

Department of Mathematics and Statistics




BGSU

Department of Mathematics

Putnam Exam
























Problem of the Month





Fall 2014



Problem 7, October 2014

A right circular cone has radius 1 and height 3. A cube is inscribed in the cone so that one face of the cube is contained in the cone. What is the side length of the cube?



Problem 6, September 2014

The integers from 1 to 10 are randomly distributed around a circle. Prove that there must be three neighbors whose sum is at least 17. What about 18? What about 19?

Correct solutions were submitted by: Linda Li and Alexandria Stough.

Please submit your solution for this month problem no later than 15 of the next month. Try to keep

your solution to less than two pages. My office is in MSC 426 and my email address is

mstaic@bgsu.edu You may also put your solution in my mailbox (Mihai Staic).

If you want to propose a problem please contact me by email or in person.



Fall 2013

Problem 5, September 2013

a) Place a knight on each square of a 4x4 chessboard. Is it possible for each knight to simultaneously make a legal move so that each knight ends up in its own square?

b) Place a knight on each square of a 7x7 chessboard. Is it possible for each knight to simultaneously make a legal move so that each knight ends up in its own square?

Spring 2013

Problem 4, April 2013

Determine all the differentiable real functions f from R to R whose composition with themselves, fof, is the original function f.

Problem suggested by Professor Sam Nadler from The University of Toledo. Correct solutions were submitted by: Nate Day and Rob Kelvey.



Fall 2012



Problem 1, August 2012

A page (two sides) is torn from a paperback novel. The sum of the remaining page numbers

is 15000. Which page was torn out? (Rasor-Bareis Exam 1996) Solution


Correct solutions were submitted by: Mike Hughes, Devin Bender, Suzanne Hirsel. There were

other students that submitted partially correct solution.

Problem 2, September 2012

Consider a bounded solid S that has the following property: Whenever we intersect S with a

plane the intersection is either the empty set or a point or a disk. Show that S must be a

sphere (a ball).



Problem 3, October 2012

Find all functions f from R to R such that f(x+y)=f(x)+f(y) and f(xy)=f(x)f(y) for all x and

y in R.