Problem of the Week #77: Monday January 20th, 2025
As before, these problems are the results of me following my curiosity, and I make no promises regarding the topics, difficulty, solvability of these problems.
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This problem is inspired by Q7 of the 1970 Canadian Math Olympiad.
In any set of 5 integers, we can always find 3 whose sum is a multiple of 3.
a). Can we find a number of integers in a set to guarantee that we can always find 2 of them whose sum is a multiple of 3?
b). Is it true that in any set of 6 integers, we can always find 4 of them whose sum is a multiple of 4?
c). What is the minimum number of integers in a set to guarantee that we can find 4 of them whose sum is a multiple of 4?
d). What’s the minimum number of integers in a set to guarantee that we can find n of them whose sum is a multiple of n?
e). One way for 3 integers to have a sum divisible by 3 is if they’re all divisible by 3, or if they all have different remainders when divided by 3. How many different remainder combinations are there for 4 integers to have a sum divisible by 4?
f). How many different remainder combinations are there for n integers to have a sum divisible by n?
g). Share your own problem inspired by this one.
h). Give one of these questions to a friend/colleague/student/family member to start a mathematical discussion.