<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN"> <html> <head> <title>Problem Solving and the Putnam Competition</title> <meta name="author" content="Andrew J. Bernoff"> </head> <body> <div align="left"> <h1><a href="http://www.math.hmc.edu/"><img alt="HMC Math" src="hmc-math4.gif" style="border: 0px solid ; width: 450px; height: 75px;"></a></h1> <h1>Problem Solving and The Putnam Competition<img alt="The Putnam Icosahedron" src="icos.gif" style="width: 155px; height: 137px;" align="right"></h1> <h2>The William Lowell Putnam Competition</h2> The Putnam Exam (officially known as the William Lowell Putnam Mathematical Competition) is a very challenging, six-hour mathematics exam taken by about 3,600 undergraduates in the United States and Canada each year. It is given on the first Saturday in December, and is composed of two sessions, one in the morning and one in the afternoon, each with six questions. Each question is worth 10 points (for a total of 120). Typically, the median score for all contestants is approximately one (out of 120). <br> <br> William Lowell Putnam, a graduate of Harvard in the class of 1882, believed strongly in the value of academic competitions. To this end, Mrs. Elizabeth Lowell Putnam, the wife of Mr. Putnam, created a trust fund for supporting such competitions. Two experimental competitions were held, one in English, and one in mathematics, before the competition assumed its present form. <br> <p>Harvey Mudd College (HMC) has a long history of participation in the Putnam, and in recent years, <a href="hmc_results.html">the school has done very well</a>. In 2011, we placed 6th in the country and in 2003, we placed 5th. We are often the only undergraduate college to place in the top ten. We also regularly sport the highest participation rate in the country with roughly 10% of our students taking the exam! <br> </p> <p>Here are the <a href="http://amc.maa.org/a-activities/a7-problems/putnam/-pdf/2011.pdf">2011 Putnam Exam Questions</a> and <a href="http://amc.maa.org/a-activities/a7-problems/putnam/-pdf/2011s.pdf">Solutions</a>.<br> </p> <h2>Harvey Mudd Putnam Links</h2> <ul> <li><a href="http://www.math.hmc.edu/rif/">The RIF Scholarship</a>- A merit scholarship offered by the <a href="http://www.math.hmc.edu/">HMC Mathematics Department</a> to students who demonstrate exceptional mathematical ability, possibly evidenced by strong performance in mathematical competitions.</li> <li>The <a href="http://www.math.hmc.edu/~su/math191/">Putnam Problem Solving Seminar</a> <!-- Andy's site http://www.math.hmc.edu/%7Eajb/m191/--> is offered every fall.</li> <li>A <a href="PutnamMHArticle.pdf">Math Horizons article</a> about the HMC Putnam Seminar.<br> </li> <li>Here are some <a href="pictures.html">pictures from the Putnam Seminar</a>.</li> <li><a href="mugs.html">Mug Shots</a> -- Pictures of the <i>beautiful</i> Putnam mugs given to all HMC Putnam participants.</li> </ul> <h2>Pizza and Problem Solving Sets from the Park City Math Institute</h2> <ul> <li>For 2012 Prof. Francis Su ran a problem session called <a href="http://www.math.hmc.edu/~ajb/PCMI/pcmi12.pdf">Symmetry</a>. </li> <li>For 2011 Prof. Andrew Bernoff ran two problem session called <a href="http://www.math.hmc.edu/~ajb/PCMI/pcmi11_a.pdf">The Sum of the Parts</a> and&nbsp;<a href="http://www.math.hmc.edu/~ajb/PCMI/pcmi11_b.pdf">Digits and Divisibility.</a> </li> <li>For 2010 Prof. Andrew Bernoff ran two problem session called <a href="http://www.math.hmc.edu/~ajb/PCMI/pcmi10_a.pdf">Induction and Recursion</a> and <a href="http://www.math.hmc.edu/~ajb/PCMI/pcmi10_b.pdf">The Extreme Principle</a>. </li> <li>For 2009 Prof. Andrew Bernoff ran two problem session called <a href="http://www.math.hmc.edu/~ajb/PCMI/pcmi09_a.pdf"> The Pigeonhole Principle</a> and <a href="http://www.math.hmc.edu/~ajb/PCMI/pcmi09_b.pdf">Parity &amp; Counting</a>. </li> <li>For 2008 Prof. Andrew Bernoff ran two problem session called <a href="http://www.math.hmc.edu/~ajb/PCMI/pcmi08_a.pdf">Hidden Symmetries</a> and <a href="http://www.math.hmc.edu/~ajb/PCMI/pcmi08_b.pdf">Lattice Points &amp; Polygons</a>. </li> <li>For 2007 Prof. Andrew Bernoff ran two problem session called <a href="http://www.math.hmc.edu/~ajb/PCMI/pcmi07_a.pdf">Some Random Problems From Probability</a> and <a href="http://www.math.hmc.edu/~ajb/PCMI/pcmi07_b.pdf">Mathematical Games</a>. </li> <li>For 2006 Prof. Francis Su ran a problem session called&nbsp;<a href="http://www.math.hmc.edu/~ajb/PCMI/pcmi06_a.pdf">Paths, Curves and Cycles</a>.&nbsp;Prof. Andrew Bernoff ran a problem session called&nbsp;<a href="http://www.math.hmc.edu/~ajb/PCMI/pcmi06_b.pdf">Digits and Divisibility.</a> </li> <li>For 2005 Prof. Andrew Bernoff ran a problem session called <a href="http://www.math.hmc.edu/~ajb/PCMI/pcmi05_a.pdf">The Sum of the Parts</a> on geometric dissection problems. Prof. Francis Su ran a problem session called <a href="http://www.math.hmc.edu/~ajb/PCMI/pcmi05_b.pdf">The Extreme Principle</a> on analyzing problems by considering extreme cases. <br> </li> <li>For 2004&nbsp; Prof. Francis Su ran two seminars, the first on <a href="http://www.math.hmc.edu/~ajb/PCMI/pcmi04_a.pdf">The Pigeonhole Principle</a> and the second called <a href="http://www.math.hmc.edu/~ajb/PCMI/pcmi04_b.pdf">Cubes and Other Geometric Objects</a>.<br> </li> <li>For 2003&nbsp; Prof. Andrew Bernoff ran two seminars one called <a href="http://www.math.hmc.edu/~ajb/PCMI/pcmi03_a.pdf">Induction and Deduction</a> and one called <a href="http://www.math.hmc.edu/~ajb/PCMI/pcmi03_b.pdf">Parity and Invariants</a>.<br> </li> </ul> <h2>Online Mathematical Resources<br> </h2> <ul> <li><a href="http://www.math.hmc.edu/funfacts/">Math Fun Facts</a> is Francis Su&rsquo;s website with many amusing math facts for students of all abilities. <br> </li> <li><a href="http://mathforum.org/students/">Math Forum</a> is a good starting place for K-12 resources.</li> <li><a href="http://www.wolframalpha.com/">Wolfram Alpha</a> is an amazingly smart cross between a calculator, a search engine and an encyclopedia.</li> <li><a href="http://mathworld.wolfram.com">Wolfram's MathWorld </a> - A very extensive encyclopedia of mathematics. Anythingthat you might want to know about math can be found here.</li> <li><a href="http://www.ams.org/mathscinet/">MathSciNet</a> - The best site for doing key word searches on journal articles.</li> <li><a href="http://oldweb.cecm.sfu.ca/projects/ISC/">The Inverse Symbolic Calculator</a> - A program that attempts to find a symbolic expression for a given decimal approximation to a number. </li> <li><a href="http://oeis.org/">Sloane's On-Line Encyclopedia of Integer Sequences</a> - An encyclopedia of many interesting sequences of integers.</li> <li>The <a href="http://www.artofproblemsolving.com/">Art of Problem Solving</a> host a set of mathematical resources, publishes books, and runs courses aimed primarily at High School students.<br> </li> </ul> <h2>Putnam Competition Sites</h2> <ul> <li><a href="http://math.scu.edu/putnam/index.html">Official Putnam Site</a> - The official rules, regulations, and results of the competition.</li> <li><a href="http://amc.maa.org/a-activities/a7-problems/putnamindex.shtml">MAA/AMC Putnam Archive</a>- An extensive archive of recent Putnam problems, solutions, and statistics. <br> </li> </ul> <h2>Putnam Preparation Web Sites</h2> <div style="margin-left: 40px;">Here is just a selection of Putnam Preparatory Seminars at various colleges. Note some of these sites move frequently.<br> </div> <ul> <li><a href="http://ocw.mit.edu/OcwWeb/Mathematics/18-S34Fall-2004/CourseHome/index.htm">MIT's Open Course Ware for Stanley &amp; Rogers Problem Solving Seminar in 2004 (18.S34)</a></li> <li>Stanley &amp; Roger's <a href="http://ocw.mit.edu/courses/mathematics/18-s34-problem-solving-seminar-fall-2007/18.s34-problem-solving-seminar-fall-2007-home-page-answer/">MIT 18.S34 Problem Solving Seminar (2007)</a><br> </li> <li><a href="http://www-math.mit.edu/%7Erstan/myputnam.pdf">Richard Stanley's Proposed Putnam Problems</a> -- from when he was on&nbsp; the committee that wrote the exam. <br> </li> <li>Ravi Vakil's <a href="http://math.stanford.edu/%7Evakil/putnam07/">Stanford Putnam Webpage</a>.<br> <a href="http://www.stats.uwaterloo.ca/%7Ecgsmall/problem.archive.html"><span style="text-decoration: underline;"></span></a></li> <li><a href="http://www.stats.uwaterloo.ca/%7Ecgsmall/problem.archive.html"><span style="text-decoration: underline;">University of Waterloo.</span></a></li> </ul> <h2>Other Online Problem Competition and Sources<br> </h2> <ul> <li><a href="http://mathforum.org/wagon/">Macalester Problem of the Week</a> - Great and offbeat&nbsp; problems due to Stan Wagon and friends !!!</li> <li>The <a href="http://amc.maa.org/a-activities/a7-problems/problemdir.shtml">MAA/AMC Website</a> has problems and solutions from other mathematics contests &nbsp;(USAMO, IMO, etc.).</li> <li>The <a href="http://www.math.vt.edu/people/plinnell/Vtregional/">Virginia Tech Regional Mathematics Competition</a> is a&nbsp; Putnam-style competition.<br> </li> <li><a href="http://www.math.purdue.edu/pow/">Purdue's Problem of the Week.</a><br> </li> <li><a href="http://www.imc-math.org/">The International Mathematics Competition</a> perhaps could be described as a worldwide version of the Putnam.<br> </li> </ul> <h2>Some Useful Problem Books</h2> Putnam level and some easier problems grouped by subject:<br> <ul> <li><span class="small"><a href="http://www.amazon.com/exec/obidos/ASIN/0387982191/ref=pd_sxp_elt_l1/t/002-7200940-4079202"><i>Problem-Solving Strategies (Problem Books in Mathematics)</i></a> &nbsp;by Arthur Engel&nbsp; </span></li> <li class="small"> <a href="http://www.amazon.com/exec/obidos/tg/detail/-/038790803X/ref=pd_sim_books_1/t/002-7200940-4079202?v=glance&amp;s=books"><i>Problem Solving Through Problems</i></a> by Loren C. Larson </li> <li class="small"> <a href="http://www.amazon.com/exec/obidos/tg/detail/-/0471135712/ref=pd_sim_books_2/t/002-7200940-4079202?v=glance&amp;s=books"><i>The Art and Craft of Problem Solving</i></a> by Paul Zeitz </li> <li class="small"><span class="small"><span class="small"> <a href="http://www.amazon.com/exec/obidos/tg/detail/-/0817641556/ref=pd_bxgy_text_1/t/002-7200940-4079202?v=glance&amp;s=books&amp;st=*"> <i>Mathematical Olympiad Challenges</i></a> by Titu Andreescu, Razvan Gelca</span></span></li> <li class="small"><a href="http://www.amazon.com/exec/obidos/ASIN/0387947434/ref=pd_pym_rvi_1/t/002-7200940-4079202"><i>Winning Solutions (Problem Books in Mathematics)</i></a> by Edward Lozansky, Cecil Rousseau</li> </ul> Collections of elementary problems appropriate for freshmen:<br> <ul> <li class="small"><span class="small"><a href="http://www.amazon.com/exec/obidos/ASIN/0883855194/ref=pd_sxp_elt_l1/t/002-7200940-4079202"><i>Five Hundred Mathematical Challenges</i></a> by Edward J. Barbeau,&nbsp;</span><span class="small"> Murray S. Klamkin, William O. J. Moser </span></li> <li class="small"><a href="http://www.amazon.com/exec/obidos/tg/detail/-/0883853256/qid=1057672535/sr=8-1/ref=sr_8_1/t/002-7200940-4079202?v=glance&amp;s=books&amp;n=507846" style="font-style: italic;">Which Way Did the Bicycle Go? : And Other Intriguing Mathematical Mysteries</a> -- Joseph D. E. Konhauser, Dan Velleman</li> </ul> Collection of Putnam probems:<br> <ul> <li class="small"><span class="small"><a href="http://www.amazon.com/exec/obidos/ASIN/088385807X/ref=pd_sxp_elt_l1/t/002-7200940-4079202"><i>The William Lowell Putnam Mathematical Competition 1985-2000</i></a><span style="font-weight: bold;"> </span>&nbsp;by Kiran S. Kedlaya et al.&nbsp;</span></li> </ul> Collections of Putnam level problems:<br> <ul> <li class="small"><span class="small"><a href="http://www.amazon.com/exec/obidos/ASIN/0486694151/ref=pd_sxp_elt_l1/t/002-7200940-4079202"><i>The Red Book of Mathematical Problems</i></a> by Kenneth S. Williams, Kenneth Hardy </span></li> <li class="small"><span class="small"><a href="http://www.amazon.com/exec/obidos/ASIN/0486695735/ref=pd_sxp_elt_l1/t/002-7200940-4079202"><i>The Green Book of Mathematical Problems</i></a> &nbsp;by Kenneth Hardy, Kenneth S. Williams&nbsp;</span></li> </ul> A classic on problem solving strategies:<br> <ul> <li class="small"><span class="small"><a href="http://www.amazon.com/exec/obidos/ASIN/0691023565/ref=pd_sxp_elt_l1/t/002-7200940-4079202"><i>How to Solve It</i></a> by George Polya</span></li> </ul> <h2>College Mathematics Journals with Problem Sections</h2> <ul> <li><a href="http://www.maa.org/mathhorizons/">Math Horizons (MAA)</a> &nbsp;is a magazine aimed at undergraduates with a very accessible problem section.</li> <li><a href="http://www.maa.org/pubs/mathmag.html">Mathematics Magazine (MAA)</a>&nbsp; is an expository journal of undergraduate mathematics, with an a good problem section for the undergraduate.</li> <li> The <a href="http://www.maa.org/pubs/cmj.html">College Mathematics Journal<i> </i>(MAA)</a> provides lively, well-motivated articles that can enrich undergraduate instruction and enhance classroom learning.</li> <li>The <a href="http://www.maa.org/pubs/monthly.html">American Mathematical Monthly</a> publishes articles about mathematics and the profession. Its readers span a broad spectrum of mathematical interests, and include professional mathematicians as well as students of mathematics at all collegiate levels. The problem section is quite challenging, and often contains unsolved problems.</li> </ul> <hr style="width: 100%; height: 2px;"><span style="color: rgb(255, 0, 0); font-weight: bold;"><span style="text-decoration: underline;">Disclaimer:</span> <span style="color: rgb(0, 0, 0);">The sites listed here at the time of the composition of this page were available openly on the web.&nbsp; I expect these sites to move and migrate over time. Moreover, much of the material referenced may be protected by copyright and other intellectual property law.&nbsp;&nbsp; The user of these resources is encourage to acknowledge the source of problems used and to obtain permissions to copy problems, solutions, resources and other documents as appropriate.</span></span><br> <hr style="width: 100%; height: 2px;"><a href="http://www.math.hmc.edu/putnam/">Putnam Homepage</a> * <a href="http://www.math.hmc.edu/">Mathematics Department </a>* <a href="http://www.hmc.edu/">Harvey Mudd College</a><br> <br> <span style="font-style: italic;">Maintained by <a href="http://www.math.hmc.edu/%7Eajb/">Andrew J. Bernoff </a></span><br style="font-style: italic;"> <span style="font-style: italic;">Last Updated on 01-08-13</span><br> </div> <br> <br> </body> </html>