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

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The discussion centers on proving that from a set of 69 distinct positive integers between 1 and 100, it is always possible to select four integers a, b, c, and d such that a + b + c = d. The proof utilizes the pigeonhole principle, starting by identifying the smallest and largest integers in the set. It then calculates the number of possible pairs for b and c within the range of 1 to 100 and compares this with the total number of integers selected. The conversation highlights the enjoyment of solving such mathematical problems and suggests further reading to enhance problem-solving skills. The conclusion affirms the existence of at least one valid quadruple within the set.
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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 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.
 
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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