1972 IMO Problems/Problem 4
Find all solutions of the system of inequalities
where
are positive real numbers.
Solution
Add the five equations together to get
Expanding and combining, we get
Every term is , so every term must
.
From the first term, we can deduce that . From the second term,
. From the third term,
. From the fourth term,
.
Therefore, is the only solution.
Borrowed from http://www.cs.cornell.edu/~asdas/imo/imo/isoln/isoln724.html