BGSU 
Department of Mathematics and Statistics 


Problem of the Month
Spring 2016
Problem 6, February 2016 Five boxes of different but unknown weights arrived in the USPS office. Mary was assigned the job of determining their respective weights. Unfortunately, all of the boxes weigh less than 100 pounds, and the scale available to him reads only weights over 100 pounds. Mary decides to weigh the boxes in pairs so that each box is weighted with every other box. The weights of all possible pairs are 110, 112, 113, 114, 115, 116, 117, 118, 120, and 121 pounds. What are the weights of the five boxes? (Solutions are due before March 1st 2016).
Problem 5, January 2016 Let A and B be points on the same branch of the hyperbola xy = 1. Suppose that P is a point lying between A and B on this hyperbola, such that the area of the triangle APB is as large as possible. Show that the region bounded by the hyperbola and the chord AP has the same area as the region bounded by the hyperbola and the chord PB.(Solutions are due before February 1st 2016). Putnam Exam, December 5^{th} 2015
Fall 2015 Problem 4, December 2015 (a) Show that for every natural number n, there exists a ndigit number that is divisible by 2^n and contains only the digits 2 and 3. (For example 2 is divisible by 2, 32 is divisible by 4, 232 is divisible by 8, etc). (b) Show that for every natural number n, there exists a ndigit number that is divisible by 5^n and contains only the digits 5, 6, 7, 8 and 9. (For example 5 is divisible by 5, 75 is divisible by 25, etc). (Solutions are due before January 1st 2016).
Problem 3, November 2015 The planar diagram below, with equilateral triangles and regular hexagons, sides length 2 cm., is folded along the dashed edges of the polygons, to create a closed surface in three dimensional Euclidean spaces. Edges on the periphery of the planar diagram are identified (or glued) with precisely one other edge on the periphery in a natural way. Thus for example, BA will be joined to QP and AC will be joined to DC. Find the volume of the threedimensional region enclosed by the resulting surface (Solutions are due before December 1st 2015).
(37th Annual Virginia Tech Regional Mathematics Contest, October 24, 2015)
Problem 2, October 2015
Problem 1, August 2015 A cake has a quadrilateral shape (four sides). We cut the cake along the two diagonals and eat one of the four pieces. The three remaining pieces weight: 120 grams, 200 grams and 300 grams respectively. What was the initial weight of the cake? (solution due before September 15^{th} 2015). Correct solution submitted by: Ben Hardy
Please submit your solution for this month problem no later than 1st 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.



