# Pi Mu Epsilon Undergraduate Math Society at USC Fall 2003 Semester

Organizers: Richard Arratia and Sergey Lototsky.

## September 8.

• Handout 1.
• A problem: n points are placed on a unit sphere. Show that the sum of the squares of distances between all points is at most n^2, with equality if and only if the center of mass of the configuration is at the center of the sphere. We solved the problem. Note that the number of dimensions of the underlying space is not important.
• A research project: describe the configurations of n identical points on a circle or a usual sphere so that the center of mass of the configuration is at the center of the circle/sphere.
• Talked a little about inequalities (arithmetic/qeometric/harmonic means; convexity).
• A problem: show that cos(x)< ((pi/2)^2-x^2)/((pi/2)^2+x^2) for 0 < x < pi/2. Note that, by the Taylor expansion, the inequality holds near the end points, but it does not mean it holds everywhere even if neither function has inflection points (a picture was presented). A more difficult inequality along these lines is in Handout 1.

## September 15.

• Property of the alternating series \sum (-1)^na_n with a_n decreasing to zero monotonically:
if L is the sum and S_n is the partial sum, then S_{2n}>L>S_{2n+1} and |L-S_n| < 1/a_{n+1}.
• Rearrangement inequality: \sum a_nb_n is the largest when both a_n and b_n are arranged in decreasing order and the smallest when one of the sequences is increasing and the other, decreasing.
• The abc problem: prove that a/(b+c)+b/(a+c)+c/(a+b)\geq 3/2 for a,b,c > 0 using the rearrangement inequality.
• Handout 2.

## September 22.

• Three solutions of the abc problem using the rearrangement inequality. All essentially used that the sum on the left is the largest possible of this kind.
• Think about problem 96B3. It is not exactly on rearrangement inequality, but in the same spirit.
• The Wallis product (giving you pi/2 as a product of even numbers over the product of odd numbers).

## September 29.

Computations using Handout 3.

## October 6.

We talked about summing powers of integers and the related topics (Bernoulli numbers and polynomials, raising and falling powers). There is a nice geometric argument why the sum of cubes is the square of the sum of the first powers.

## October 13.

• Handout 4.
• We discussed a problem from Handout 4: why x^4+y^4=z^2 has no integer solutions.
• We proved the representation for the Pithagorian triples: if a,b,c are relatively prime and a^2+b^2=c^2, then, necessarily, exactly one of a or b is even (say, b) and a=m^2-n^2, b=2mn, c=m^2+n^2.

## October 20.

We talked in general about arithmetic modulo m.

## October 27.

• We proved that an odd prime p can be written as sum of two squares if and only if p=4k+1.

## November 3.

• Definition and properties of the Euler function.
• Discussion of Problem 8 from Handout 4 (that the sequence a_1=2, a_{n+1}=2^{a_n} becomes eventually constant mod m for every integer m> 0.)

## November 10.

• Some unfinished business about the Euler theorem and the problem with growing tower of powers.
• Problem 5 from Handout 5.

## November 17.

• Problems 1-4 from Handout 5. In particular, three possible solutions for problem 3 (by generating functions, by Chu-Vandermond convolution, and directly) and two possible solutions for problem 4 (directly and by identifying the integer first).

## November 24.

• Problem 5 from Handout 5 revisited.
• Problem B-5 from the 2002 Putnam exam.
• Food for thought: what interesting properties of the number 2003 can you think of?