Problems and Contests (Math 394) --- Fall 2012
Do you enjoy solving problems? Do you like contests?

The undergraduate mathematics seminar on Problems and Contests may be for you. Come and participate when you can. For the Fall 2012 semester, we meet
Wednesday,   4:10 – 5:00 pm,   in Math 306

Here is our second set of problems for the Fall 2012 semester.
  1. Prove: If the vertices of a trapezoid lie on a circle, then its diagonals have the same length.
    trapezoid in a circle             trapezoid in a circle             trapezoid in a circle
    Click on a figure to display an interactive webpage which lets you explore by moving points A, B, C.
    This was problem 5 of our Lennes Competition in 2010.
  2. Suppose A and B are complex numbers such that   |A| = 1 = |B|   and   A · B ≠ −1.  
    Prove that   (A + B) / (1 + A · B)   is a real number.
  3. Find a positive number such that one-fifth of it multiplied by one-seventh of it equals the number.
    Is there a negative number with the same property?
  4. A standard Bridge deck of 52 playing cards is shuffled and placed facedown on a table; then the cards are shown one after another (dealing from the top). Suppose you are allowed to bet (in advance) when the first black Ace will appear. What number should you pick to maximize your long-run success if this game will be repeated many times?
  5. Find all integers   n   such than   1 + n + n2 + n3 + n4   is the square of an integer.
    ASCM Problem Solving Competition, September 2012

Last modified: 7 November 2012, Wednesday 10:42