Putnam Mathematical Competition

The 70^th annual William Lowell Putnam Mathematical Competition --- a competitive examination in collegiate mathematics --- will be given Saturday, December 5, 2009 to undergraduate contestants from colleges and universities in the United States and Canada.

At the University of Montana, our competition location is Math 108 and the times are 8:30–11:30 AM and 1:30–4:30 PM. (If an afternoon football playoff game is alluring, participating in just the morning session would be fine.)

The organization of the exam is described in a website at Santa Clara University math.scu.edu/putnam The document math.scu.edu/putnam/describtcJan.html describes the exam and discusses its scope & level of difficulty.
The Putnam Competition is administered by The Mathematical Association of America (www.maa.org).
David Rusin has a website www.math.niu.edu/~rusin/problems-math with many links to collections of problems (and solutions) from old exams.
A collection of recent exams, together with some solutions, amc.maa.org/a-activities/a7-problems/putnamindex.shtml is published by AMC (American Mathematics Competitions).

Here are several problems from recent Putnam exams.

Let S be a set of real numbers which is closed under multiplication (that is, if a and b are in S, then so is a*b). Let T and U be disjoint subsets of S whose union is S. Given that the product of any three (not necessarily distinct) elements of T is in T and the product of any three elements of U is in U, show that at least one of the two subsets T, U is closed under multiplication. (problem A1, 1995)
Suppose that each of twenty students has made a choice of anywhere from zero to six courses from a total of six courses offered. Prove or disprove: There are five students and two courses such that all five have chosen both courses or all five have chosen neither. (problem A3, 1996)
Given a point (a,b) with 0 < b < a, determine the minimum perimeter of a triangle with one vertex at (a,b), one on the x-axis, and one on the line y = x. You may assume that a triangle of minimum perimeter exists. (problem B2, 1998)
Prove that there exist infinitely many integers n such that n, n+1, and n+2 are each the sum of two squares of integers. [Example: 0 = 0² + 0², 1 = 0² + 1², and 2 = 1² + 1².] (problem A2, 2000)
Can an arc of a parabola inside a circle of radius 1 have length greater than 4? (problem A6, 2001)
For which real numbers c is there a straight line that intersects the curve y = x⁴ + 9 x³ + c x² + 9 x + 4 in four distinct points? (problem B2, 1994)

Last modified: 18 November 2010, Thursday 09:28