## Lennes Awards in Mathematics

 what 2017 N. J. Lennes Award Competition in Mathematics when Thursday, 6 April 2017 — start between 2 and 4 pm, work for two hours. where Room 306 in the Mathematics Building (computers in this room may be used during the competition) why Gain experience in mathematics contests. Scholarship Prizes will be awarded. how On the basis of an examination in undergraduate mathematics A computer or calculator (or both) may be used during the competition. who Any UM undergraduate. Students currently taking calculus are encouraged to participate. register Sign up in the Math Office (room 102 in Mathematics Building, phone 243-5311).

The University of Montana Department of Mathematical Sciences awards prizes (on the basis of the Lennes Contest Exam) each year to undergraduates in memory of Nels Johann Lennes (1874-1951), former Professor of Mathematics at The University of Montana. He is remembered as the chairman of Mathematics for over thirty years until retirement in 1944. The house he built on Gerald Avenue, one of the "mansions of Missoula", is now the residence of the University President. As an educator, he wrote over one hundred textbooks. In the mathematical community, Lennes is best remembered (along with F. Hausdorff) for developing the fundamental concept of a "connected topological space". (The article Connected Sets and the AMS, 1901–1921 (AMS Notices, April 2009, vol 56 pp 450–458) discusses some work by NJ Lennes; its author, David Zitarelli, also gave a colloquium presentation here in April 2010.)

Lennes Competition   —   sample problems
1. A quart of low-fat milk is two percent butterfat. This is advertised as thirty-eight percent less fat than whole milk. What should be the advertised reduction for one percent butterfat?
2. The diagonals of a trapezoid divide it into four triangles. Two of these triangles have parallel sides --- let their areas be A and B. Find the area of the trapezoid in terms of A and B.
3. A circular pie can be sliced into two pieces with one straight cut. A second cut that crosses the first cut will produce a pie sliced into four pieces.
1. How many pieces can be obtained with a third cut?
2. What is the largest number of pieces that can be obtained with four cuts?
3. Suppose n is a positive integer. What is the largest number of pieces obtainable with n cuts?
4. Prove that a parcel sufficiently large can be mailed by using only 7-cent and 13-cent stamps. How large is large enough?
5. Two students take turns eating a loaf of bread. Each always eats one-fifth of the remaining part of the loaf. How much does each student eat?
6. Find non-constant differentiable functions p and q which satisfy the
Naive Product Rule:       (p · q)'   =   p' · q'
(i.e., derivative of product equals product of derivatives).

Several recent Lennes Competitions are published online at the URL
www.math.umt.edu/lennes/exams.html

Last modified: 3 March 2017, Friday 14:05