1963 IMO Problems/Problem 4
Problem
Find all solutions of the system where is a parameter.
Video Solution
https://youtu.be/n02OTRpHO20?si=p7WSt-NZNU-LsAPe [Video Solution by little-fermat]
Solution
Notice: The following words are Chinese.
首先,我们可以将以上5个方程相加,得到:
当时,因为关于原方程组轮换对称,所以
若反之,则方程两边同除以,得到,显然解为
综上所述,若,最终答案为,否则答案为
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, .
Mistake
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).
See Also
1963 IMO (Problems) • Resources | ||
Preceded by Problem 3 |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 5 |
All IMO Problems and Solutions |