BGSU Department of Mathematics and Statistics BGSU Department of Mathematics Putnam Exam 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 1-st 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 1-st 2016). Putnam Exam, December 5th 2015 Fall 2015 Problem 4, December 2015 (a) Show that for every natural number n, there exists a n-digit 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 n-digit 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 1-st 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 three-dimensional region enclosed by the resulting surface (Solutions are due before December 1-st 2015). (37th Annual Virginia Tech Regional Mathematics Contest, October 24, 2015) Problem 2, October 2015 Three men in a cafe order a meal the total cost of which is \$15. They each contribute \$5. The waiter takes the money to the chef who recognizes the three as friends and asks the waiter to return \$5 to the men.The waiter is not only poor at mathematics but dishonest and instead of going to the trouble of splitting the \$5 between the three he simply gives them \$1 each and pockets the remaining \$2 for himself.Now, each of the men effectively paid \$4, the total paid is therefore \$12. Add the \$2 in the waiters pocket and this comes to \$14.....where has the other \$1 gone from the original \$15? (solution due before November 1st 2015). Correct solution submitted by: Peter W. Barclay 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 15th 2015). Correct solution submitted by: Ben Hardy Please submit your solution for this month problem no later than 1-st 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 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.