Math club
I once had a wee math club which met up to once a week to discuss such
problems and curiosities as caught our interest. I typed up notes on what
we talked about; these are they. Be warned: these notes contain errors
(though probably not more than six or seven each).
- Future topics
- A list of problems and topics we might look at in the future.
- Math Club Notes: 2007 August 22
- The cubes mod p, with Cauchy's theorem.
A ring in which every uninvertible element gives rise to infinitely many
congruence classes.
The determinant of a Vandermonde matrix.
- Math Club Notes: 2007 May 30
- Polynomials that commute with squaring.
A trigonometric inequality, two ways.
A telescoping trigonometric product.
An arcsin identity.
More rings that don't pin coefficients.
- Math Club Notes: 2007 April 4
- Partial results on the cubes mod p.
- Math Club Notes: 2007 March 7
- Using Vandermonde matrices to show that fields pin coefficients.
A weird problem from Barbeau.
- Math Club Notes: 2007 February 28
- How many multiplications it takes to evaluate a determinant.
A preliminary observation on "coefficient pinning" rings.
Two equivalent but rather different-seeming statements.
- Math Club Notes: 2007 February 16
- A simple result in topological linear spaces.
- Math Club Notes: 2007 January 24
- Matrix sum equal to matrix product.
Expressing open sets as preimages of open sets.
Polynomials with integer values for integer arguments.
Erroneous argument involving the tangent of a sum.
- Math Club Notes: 2007 January 10
- Archimedes' weird bisection lemma.
Area of an average rectangle.
Dodgson's problem on sums of squares.
The sum and product of the tangents of a triangle's angles.
Reasons some functions aren't polynomials.
- Math Club Notes: 2006 November 27
- A construction with straightedge alone.
Some reasons the sine function isn't a polynomial.
Why disjunction can't be expressed using negation and xor.
- Math Club Notes: 2006 November 20
- A proof of the rearrangement inequality.
Hint on Archimedes' weird segment bisection lemma.
Making negation using other boolean operations.
- Math Club Notes: 2006 October 30
- The "mediant inequality" and the mean value theorem,
applied to a broken contest problem from Waterloo.
- Math Club Notes: 2006 May 23
- Boolean rings, again.
Catalan's identity.
A couple matrix problems.
- Math Club Notes: 2006 May 16
- A broken probability argument.
Rusin's exam question about matrix traces.
Counting is better than case analysis (three examples).
Questions from a podcast.
- Math Club Notes: 2006 May 2
- An easy estimate on the sum of the reciprocals of the squares.
Integrating secant.
- Math Club Notes: 2006 March 30
- A triangle problem by Charles Dodgson.
Geometric interpretation of a parameterization of the unit circle.
Some trivial trigonometric inequalities done the hard way.
Euclid's algorithm in terms of linear algebra.
Problem roundup.
- Math Club Notes: 2006 February 9
- An inequality, solved. Problem roundup.
- Math Club Notes: 2006 January 19
- A trigonometric/hyperbolic identity.
Proving concurrency.
Expressing logical or in terms of other operations.
An inequality.
- Math Club Notes: 2006 January 12
- Alternative methods for past problems:
the sum of every third binomial coefficient,
and a continued square root with twos.
An algebraic identity concerning circles.
- Math Club Notes: 2005 December 8
- Two of the 2005 Putnam problems:
finding a polynomial which vanishes for certain kinds of argument (B1),
and an integral (A5).
- Math Club Notes: 2005 December 1
- Some old Putnam problems:
a series with a parameter, rational for some values, irrational for others;
a sequence containing all positive rationals;
a certain sequence of integers is eventually periodic;
finding the minimum of a scary-looking expression;
simplifying an eighth root of a continued fraction;
extracting one recurrence relation from another.
- Math Club Notes: 2005 November 3
- A Putnam problem (2003 A2): the sum of geometric means
is at most the geometric mean of the sums.
- Math Club Notes: 2005 October 27
- Sum of every third binomial coefficient: some closed forms.
- Math Club Notes: 2005 October 13
- A solution to a U of Waterloo contest question:
functions with an unusual property.
Link to someone's solutions of the 2005 IMO problems.
- Math Club Notes: 2005 September 22
- Spurious solutions.
A calculus for the maximum function.
Numbers congruent to their own squares.
- Math Club Outstanding Problems: 2005 September 11
- A list of our outstanding problems.
- Math Club Notes: 2005 August 29
- Solutions to two more 2005 IMO problems:
numbers relatively prime to all numbers of the form 2^n + 3^n + 6^n - 1;
a certain sequence is an enumeration of the integers.
- Math Club Notes: 2005 August 22
- 2005 IMO problems:
equilateral hexagon in equilateral triangle;
combinatorics with a 6-problem contest.
- Math Club Notes: 2005 August 8
- Integer-sided rectangles with equal area and perimeter.
Convergence of integral of ln sin x.
Hint on the sum of every third binomial coefficient.
A new problem: finding all functions with an unusual property.
- Math Club Notes: 2005 July 18
- Hyperbola tangents.
- Math Club Notes: 2005 July 11
- Continued square roots.
Examples of half-lattices.
Solving for unknown quantities vs solving for unknown algebraic structures.
Sum of reciprocals of sums of squares.
- Math Club Notes: 2005 July 4
- An integral.
A sum.
- Math Club Notes: 2005 June 27
- Counting compositions.
Stuff by Timothy Gowers.
A few sums.
Empty sums and products.
- Math Club Notes: 2005 June 20
- Bitstrings with no consecutive zeroes.
Combinatorics of distributivity: two simple applications.
- Math Club Notes: 2005 June 13
- An integral.
Examples of a simple proof-finding strategy:
circular relations; the Cauchy Mean Value Theorem.
Famous series by integrating the geometric series.
The integers, bounded above.
- Math Club Notes: 2005 June 6
- Sum of squared distances on a sphere.
Solution for triangle decomposition problem.
Upper bounds on the integers.
The converse of the Intermediate Value Theorem.
- Math Club Notes: 2005 May 30
- A Putnam problem about triangle decomposition.
Application of the Mean Value Theorem.
Thoughts about how to study math.
A combinatoric view of distributivity.
- Math Club Notes: 2005 May 22
- Web sites.
Proofs of the Cauchy-Schwarz inequality.
The Bernoulli inequality.
Deriving Binet's formula using linear algebra.
Area of a parallelogram.
Telescoping sums (and products).
- Math Club Notes: 2005 May 16
- Geometric vs algebraic proofs.
Two proofs that positive numbers have nth roots.
Application of the Cauchy-Schwarz inequality.
- Math Club Notes: 2005 May 9
- Truth table for implication.
Perturbing sums to find closed forms.
Ordered fields.
The arithmetic-geometric mean inequality.
- Math Club Notes: 2005 May 2
- Why Boolean rings are so called.
Some useful isomorphisms.
Tangents to a parabola.
Mail Steven: steven@amotlpaa.org