Problem of the Week #74: Monday December 2nd, 2024
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 Canadian Math Olympiad 1970 Q1
A set of 3 real numbers satisfies the symmetry that any number added to the product of the other 2 will result in 2. There are 2 such sets: all 1s and all -2s.
a). How many such sets are there if any number in the set added to the product of the other 2 results in 1? 0?
b). What’s the maximum number of 3-element sets that satisfy the symmetry rule if each element added to the product of the other 2 results in an integer n? Real number n?
c). For which real number values of n in the symmetry rules do the 3-element sets have to contain 3 identical numbers?
d). A set of 4 real numbers satisfies the symmetry that any number added to the product of the other 3 will result in 5. How many such sets are there?
e). A set of 4 real numbers satisfies the symmetry that the sum of the pair-wise products of the elements will be 5. How many such sets are there?
f). What symmetry rules should we place on a set of n real numbers to get the most number of sets satisfying the rule?
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.