Difference between revisions of "1972 IMO Problems/Problem 4"
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==Solution== | ==Solution== | ||
− | Add the five | + | Add the five inequalities together to get |
<math>(x_1^2 - x_3 x_5)(x_2^2 - x_3 x_5) + (x_2^2 - x_4 x_1)(x_3^2 - x_4 x_1) + (x_3^2 - x_5 x_2)(x_4^2 - x_5 x_2) +</math> | <math>(x_1^2 - x_3 x_5)(x_2^2 - x_3 x_5) + (x_2^2 - x_4 x_1)(x_3^2 - x_4 x_1) + (x_3^2 - x_5 x_2)(x_4^2 - x_5 x_2) +</math> |
Revision as of 13:36, 29 April 2025
Find all solutions of the system of inequalities
where
are positive real numbers.
Solution
Add the five inequalities together to get
Expanding, multiplying by , and re-combining terms, 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 [1]
See Also
1972 IMO (Problems) • Resources | ||
Preceded by Problem 3 |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 5 |
All IMO Problems and Solutions |