1990 IMO Problems/Problem 6
Prove that there exists a convex with the following two properties: (a) All angles are equal. (b) The lengths of the sides are the numbers in some order.
Let and . Then the problem is equivalent to that there exists a way to assign such that . Note that , then if we can find a sequence such that and , the problem will be solved. Note that , then can be written as a sum from two elements in sets and . If we assign the elements in in the way that , then clearly Similarly, we could assign elements in in that way () to . Then we make according to the previous steps. Let (mod 5), (mod 199), and , then each will be some and And we are done.
This solution was posted and copyrighted by YCHU. The original thread for this problem can be found here: 
Throughout this solution, denotes a primitive th root of unity.
We first commit to placing and on opposite sides, and on opposite sides, etc. Since , , , etc., this means the desired conclusion is equivalent tobeing true for some permutation of .
Define . Then notice that and so summing yields the desired conclusion, as the left-hand side becomesand the right-hand side is the desired expression.
This solution was posted and copyrighted by v_Enhance. The original thread for this problem can be found here: 
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