Math Contest

Math Department contest No. 2.1

October, 1997


RULES: ALL are encouraged to participate. Submit your solutions to any or all of the problems, in the box located in the math department conference room by NOV 4th. Be sure to SHOW ALL WORK. Undergraduates who submit correct solutions will get prizes which will be given out at the next coffee hour NOV 5th. Be sure to give your name and say if you are an undergraduate or not.


  1. (Submitted by Mary Ann Saadi) Let tex2html_wrap_inline30 be an increasing sequence of positive integers such that
    displaymath28
    If tex2html_wrap_inline32, find tex2html_wrap_inline34.
  2. (Submitted by Gerry Ladas) In a large urn there are 75 white balls and 150 black ones, and beside the urn is a big pile of black balls. Now, the following two-step operation is performed repeatedly. First, two balls are withdrawn at random from the urn and then

    1. if they are both black, one of them is put back and the other is thrown away.
    2. if one is black and the other white, the white one is put back and the black one is thrown away,
    3. if they are both white, they are both thrown away and a black ball from the pile is put into the urn.

    Therefore, whatever the case, at each stage two balls are removed from the urn and only one is put back, thus reducing the number of balls in the urn by one. Eventually, then, the urn will reach the point of containing just a single ball. The question is ``What color is this last ball?"

  3. (Austrian-Polish High School Math Competition, 1978) Consider 1978 sets, tex2html_wrap_inline36, each containing 40 elements. If each two of the sets intersect in exactly one element, prove that there is an element t that is common to all 1978 of the sets.
  4. (Submitted by Jim Lewis) Each of the numbered groups of bars corresponds to a zip code below. The Postal Service uses this system in automated mail processing: to guarantee accuracy, an additional number, called a correction character, is added to each of the codes to make the sum of its digits a multiple of 10 (for example, 12345 becomes 123455). Match each bar with its corresponding zip code. (NOTE: THERE IS A FIGURE THAT GOES WITH THIS QUESTION THAT IS NOT AVAILABLE HERE. ASK YOUR MATHEMATICS INSTRUCTOR FOR A COPY)

Prof. Nancy Eaton
Wed Oct 15 16:13:12 EDT 1997