Problems of the Month - December 2006

We would like to receive your solutions and comments. Our intention is to post a digest of the received material (plus our own comments) together with the next "problem of the month" (in the present instance, owing to summer vacation, this will probably not be before the beginning of the Fall term).

Solutions can be sent to shapiro@math.kth.se

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  1. Suppose that a convex domain D in the plane with smooth boundary has the following property: There is a point P in D such that for every arc J contained in the boundary of D, the harmonic measure of J as seen from P, equals the angle which J subtends, as seen from P, divided by 2π. Show that D must be a circle and that the point P is the center.

  2. Show there is a convex domain D in the plane with smooth boundary and DISTINCT points P, Q in D for which the following is true:
    For every arc J contained in the boundary of D, the harmonic measure of J as seen from P, equals the angle which J subtends as seen from Q divided by 2π.

    The questions 3a) and 3b) are UNSOLVED:

  3. a) Describe all such domains.
    b) Is there an analogous example of a convex domain in R3 ?

    Problem 2 can also be generalized as follows:
  4. Characterize all simply connected domains D having the property that there exist finitely many points P1,...,Pn and Q in D such that for every arc J on the boundary of D the angular measure of J as seen from Q (divided by 2π) equals a linear combination (independent of J) of the harmonic measures of J at the points P1,...,Pn
    The characterization can be formulated for example in terms of a conformal map (one-to-one analytic function) from the unit disk onto D. (This problem is a kind of miniresearch which can be further developed in different directions.)

    Remark: If D is an open, bounded domain in the plane with boundary ∂D the harmonic measure of an arc J in ∂D at a point P of D is defined as the value at P of the solution u(x,y) to Laplace's equation Δu = 0, with boundary condition u = 1 on J and u = 0 otherwise on ∂D.


    Here is another, perhaps more elementary algebraic problem:

  5. A Hadamard matrix is, by definition, a matrix whose entries are from the set {-1,1} and whose rows are mutually orthogonal. Suppose we have an n x n Hadamard matrix and the upper left
    p x q section has all entries +1. Prove that pq cannot exceed n.



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Problem of the Month - June 2006

The classical soccer ball, introduced at the world championship 1970, consists of pentagonal and hexagonal leather pieces sewn together along the edges, see picture. Each hexagon is surrounded by alternatively pentagons and hexagons, and each pentagon is surrounded by only hexagons. At each corner exactly three leather pieces meet.

Show that with this pattern there is only one possible size of the ball, counted as the number of pentagons and hexagons. Also find these numbers.

The following "ball" is composed of only hexagons. Show that this ball is in fact impossible.

Problem contributed by Björn Gustafsson and Torbjörn Odelman

Solution