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Math olympiad problem

  1. May 17, 2005 #1
    i have 69 distinct positive whole numbers between 1 and 100. i pick out 4 integers a,b,c,d. prove that i can always pick out 4 integers such that a+b+c=d. can this always hold true with 68 positive integers?
  2. jcsd
  3. May 17, 2005 #2
    i'm assuming that you don't have to pick out a+b+c=d all the time but there exists a quadruple in your set of 69.

    The proof relies on teh pigeon hole principle:
    [1]let a and d be the smallest and largest of your set of 69 respectively.
    [2]find the number of pairs taht b & c can be within 1-100 and the set of 69.
    [3]Compare these two values.
  4. May 18, 2005 #3
    thanks! I seem to know what to do now. The fun part about this problem is the pigeon hole principle. I love math problems like this one.
    one of my favorite math problems (though simple) is this:
    Prove that at any party, two friends at that party must have the same amount of friends present there (given ofcourse that if someone is a friend to you, you are friend to that someone).
  5. May 18, 2005 #4
    if you need a good book to further your proving skillz pick up
    "problem solving" by Larson.
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