Math Olympiad Problem: Proving a+b+c=d with 69 Distinct Integers between 1-100

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Discussion Overview

The discussion revolves around a mathematical problem involving the selection of four integers from a set of 69 distinct positive whole numbers between 1 and 100, specifically proving that there exist integers a, b, c, and d such that a + b + c = d. The scope includes theoretical reasoning and mathematical proof techniques.

Discussion Character

  • Exploratory, Technical explanation, Mathematical reasoning

Main Points Raised

  • One participant proposes that it is possible to always find four integers a, b, c, and d such that a + b + c = d within a set of 69 distinct integers.
  • Another participant suggests that the proof can be approached using the pigeonhole principle, outlining steps to compare the number of pairs that can be formed with the integers.
  • A different participant expresses enthusiasm for the problem and mentions a related mathematical problem about friends at a party having the same number of friends.
  • One participant recommends a book titled "Problem Solving" by Larson to enhance proving skills.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the proof's validity or the necessity of the condition regarding the number of integers. Multiple viewpoints and approaches are presented without resolution.

Contextual Notes

The discussion does not clarify the specific assumptions or definitions used in the proof, nor does it resolve the mathematical steps involved in the proposed proof using the pigeonhole principle.

tongos
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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?
 
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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 the 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.
 
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).
 
if you need a good book to further your proving skillz pick up
"problem solving" by Larson.
 

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