1963 IMO Problems/Problem 4
Find all solutions of the system where is a parameter.
https://youtu.be/n02OTRpHO20?si=p7WSt-NZNU-LsAPe [Video Solution by little-fermat]
Notice: The following words are Chinese.
The solution in English (translated by Google Translate):
First of all, we can add the five equations to get:
When , Because is symmetric in the original equations,
Otherwise, dividing both sides by , we get , and clearly
Summarizing, if , then the answer is of the form . Otherwise, .
While doing this question, I found out that the answer is actually wrong, can equal and and still produce an infinite number of solutions in the form where is a real number and the set is cyclic (Ex: The set can correspond to or , either works. Order matters, but not starting position.). For example, if and the set will be , which you can test and find out that it still works even though the set isn't symmetric.
Can someone change this answer so it's correct?
Edit: 亲爱的中国盆友，我找到错误了。 If , y can be anything(不一定要轮换对称).
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