1963 IMO Problems/Problem 4
Problem
Find all solutions of the system
where
is a parameter.
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,
.
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 |