The William Lowell Putnam Competition is the premier university math problem solving competition in North America. The participants compete individually and universities may designate three individuals for the team competition. There are cash prizes for top scorers and the 500 highest ranked individuals are named in a list that is sent to graduate schools (only the top 100 individuals and top 10 schools are named publicly). Memorial University has a strong record in the competition, having ranked among the top 100 universities in North America seven times in the last ten years:

2005 Andrew Critch, Neil McKay, Justin Rowsell team rank 35

2007 Andrew Stewart, Jason Gedge, Bradley Dart team rank 76

2008 Andrew Stewart, Jason Gedge, Ian Payne team rank 43

2010 Jonathan Lomond, Ian Payne, Mark Penney team rank 47

2012 Jonathan Lomond, Brandon Thorne, Matthew Sullivan team rank 73

2013 Brandon Thorne, Chris Pardy, Adam Gardner team rank 95

2014 Brandon Thorne, Chris Pardy, Noah MacAulay team rank 44

The competition is written on the first Saturday in December and consists of two 3 hour sessions: Part A in the morning and Part B after lunch. Each part consists of 6 problems, making 12 problems in total. Most people who write the Putnam don’t solve any problems, so getting even one correct is an accomplishment. You can many old exams here, a sample of practice problems here, and more training resources here.

The Putnam problems are numbered roughly in order of difficulty.

- Questions 1 and 2 are usually the least difficult but they still quite hard (solved by 80% and 65% of top 200 respectively),
- Questions 3, 4 and 5 tend to be a higher level of difficulty (solved by 40%, 35% and 20% of top 200 respectively),
- Questions 6 is usually too hard to bother with (solved by only about 6% of top 200).

A smart strategy for writing the Putnam is to focus mainly on Questions 1 and 2. If you successfully solve one and can’t make progress with the other, you should consider moving on to Questions 3, 4 and maybe 5. Question 6 should be skipped by all but the most elite problem solvers.

Students who are interested in taking part in the Putnam should contact me or Margo Kondratieva to get included on the mailing list. Practice sessions and internal contests with cash prizes take place throughout the year.

**Preparing for the Putnam**

Most Putnam problems can be solved using mathematics at the 2000 course level or below, though 3000 level material is often useful. Here is a range of topics it would be helpful to review in preparation for the exam.

DISCRETE MATH (Math 2320, 3340, 3370, Stats 3410)

Logic and Sets: Proof by induction, proving the contrapositive, proof by contradiction, the pigeonhole prince, functions, onto-to-one and onto.

Integers: Modular arithmetic, gcd and lcm, prime factorization, well ordering principal and induction, fractions with relatively prime numerator and denominator, Z_n as a ring and Z_p as a field, polynomials over Z_p^r, greatest integer part.

Combinatorics: Permutations, factorials, product sets, binomial coefficients, geometric series and other series, principle of inclusion-exclusion, roots and symmetric polynomials.

Probability: probability distributions, the mean, uniform distribution, expected value, independent random variables.

CALCULUS (Math 1000, 1001, 2000, 3202, 3210)

Calculus in one variable: differentiation, tangent lines, change of variables, implicit differentiation, FTC, IVT, norm inequality for integrals, trig identities, even/odd functions, rational functions and long division, inequalities, polynomials, quadratic formula, logarithms, power series, limits, integral approximation of series.

Multivariable calculus: parametrized curves, cross product for area, change of variables (using symmetry), polar and cylindrical coordinates, convexity and the convex hull, IVT, linearity of integration, max-min on closed regions, polynomials, gradients and path integrals.

Complex numbers, polynomials and functions: Difference of squares, complex roots, roots of unity, complex conjugation, multiple roots and polls, f'(x)/ f(x), harmonic functions.

Geometry: Triangles, law of sines and cosines, circles, lines, collinearity, great circles, Euler characteristic formula for simply connected polyhedra V-E+F=2.

ALGEBRA (Math 2050, 2051, 3320)

Linear algebra: Linear equations, determinants and row operations, linear maps, rank-nullity theorem, linear dependence, orthogonal transformations, symmetric matrices and quadric forms, eigenvalues, the dot product, linear algebra over the rationals, summation notation.

Finite groups: Lagrange’s Theorem and orbit-stabilizer, concept of centralizer, finite abelian groups, Z_n as a ring.

## 0 Comments