Week of 1/20

In the comments, please suggest some interesting problems for the week of 1/20.

14 thoughts on “Week of 1/20

  1. We say a line L on a plane “cuts” a set of points A, if on both sides of L there are points from A. Prove that if the points z_1, z_2, …, z_n \in \mathbb{C} aren’t all on the same line and z_1+z_2+ …+ z_n=0 then every line passing through the origin cuts {z_1, z_2, …, z_n}.

    (I probably messed up the Latex, so someone will have to fix it up).

    • Screw this. Here we go again:

      We say a line L on a plane “cuts” a set of points A, if on both sides of L there are points from A. Prove that if the points z_1, z_2, …, z_n (represented as complex numbers) aren’t all on the same line and z_1+z_2+ …+ z_n=0 then every line passing through the origin cuts {z_1, z_2, …, z_n}.

  2. This one requires slightly more knowledge:

    Suppose p is a prime number. Let H, K be subgroups of G, such that H is a p-group and K is normal in G with index [G : K] coprime to p. Show H is contained in K. (Difficulty: Standard)

  3. There are 650 points inside a circle of radius 16. Prove that there exists a ring with inner radius 2 and
    outer radius 3 covering at least ten of these points.

  4. An auditorium has a rectangular array of chairs. There are exactly 14 boys seated in each row and exactly 10 girls seated in each column. If exactly 3 chairs are empty, prove that there are at least 567 chairs in the auditorium.

  5. What happened to the problem of the week???????????????????????????????????????????????????????????????????????????????????????????????????????????

  6. Although the week of 1/20 has come and gone, and no update has happened since, here’s a few of suggestions:

    1. Give an example of a group that is the union of three proper subgroups.

    2. Find the Lebesgue measure of the Cantor middle [; \frac{1}{n} ;] set for odd [; n>1 ;]. That is, construct a set similar to the Cantor middle thirds set by taking away the middle [; \frac{1}{n} ;] rather than the middle thirds. Show that these Cantor sets are all homoeomorphic as subspaces of [; \mathbb{R} ;].

    3. Show that, for every [; n\geq1 ;], there is a quotient space of the Cantor set homeomorphic to the closed unit [; n ;]-cube, [; I^{n} ;].

    4. For any real [; n\times n ;] matrix [; A ;], define [; A_{k} ;] to be the top left [; k\times k ;] submatrix for [; 1\leq k\leq n ;]. Show that a real [; n\times n ;] matrix [; A ;] is positive definite iff [; \det\left(A_{k}\right)>0 ;] for [; 1\leq k\leq n ;].

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