1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Putnam 2000 b1

  1. Nov 10, 2007 #1
    1. The problem statement, all variables and given/known data

    Let a_j,b_j,c_j be integers for 1 <= j <= N. Assume, for each j, at least one of a_j, b_j, c_j is odd. Show that there exist integers r,s,t s.t. r a_j + s b_j +t c_j is odd for at least 4N/7 value of j.

    Let 0 represent even numbers and 1 represent odd numbers since everything is mod 2.

    We can put each ordered triple (a_j, b_j, c_j) in one of the 7 bins: (1,1,1) (1,1,0) (1,0,1) (1,0,0) (0,1,1) (0,1,0) (0,0,1)

    Now I can prove that some set of 4 of those bins must contain 4N/7 ordered pairs. We need only prove that, given a set of 4 of those bins, we can find r,s,t that makes those 4 bins odd. Does anyone know how to do that? Is that a good approach? Will that work?


    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Nov 10, 2007 #2
    Does my attempted solution make sense to people?
     
  4. Nov 11, 2007 #3
    If you go here: http://www.unl.edu/amc/a-activities/a7-problems/putnam/-pdf/2000s.pdf
    and look at the solution, does anyone get else get confused near then end?

    In particular, shouldn't it be "exactly four of the seven" instead of "at least four of the seven" in the third sentence? And in the fourth sentence, shouldn't that be exactly instead of at least?
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Putnam 2000 b1
  1. Putnam 1951 A6 (Replies: 3)

  2. Putnam Problem 2010 A5 (Replies: 1)

Loading...