Difference between revisions of "1972 IMO Problems/Problem 4"
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(x_5^2 - x_2x_4)(x_1^2 - x_2x_4) \leq 0</cmath> | (x_5^2 - x_2x_4)(x_1^2 - x_2x_4) \leq 0</cmath> | ||
where <math>x_1, x_2, x_3, x_4, x_5</math> are positive real numbers. | where <math>x_1, x_2, x_3, x_4, x_5</math> are positive real numbers. | ||
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==Solution== | ==Solution== |
Revision as of 15:34, 17 October 2014
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.