Difference between revisions of "1969 IMO Problems/Problem 6"
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==Problem== | ==Problem== | ||
| − | Prove that for all real numbers <math>x_1, x_2, y_1, y_2, z_1, z_2</math>, with <math>x_1 > 0, x_2 > 0, x_1y_1 - z_1^2 > 0, x_2y_2 - z_2^2 > 0</math>, the inequality<cmath>\frac{8}{(x_1 + x_2)(y_1 + y_2) - (z_1 + z_2)^2} \leq \frac{1}{x_1y_1 - z_1^2} + \frac{1}{x_2y_2 - z_2^2}</cmath>is satisfied. Give necessary and sufficient conditions for equality. | + | Prove that for all real numbers <math>x_1, x_2, y_1, y_2, z_1, z_2</math>, with <math>x_1 > 0, x_2 > 0, y_1 > 0, y_2 > 0, z_1 > 0, z_2 > 0, x_1y_1 - z_1^2 > 0, x_2y_2 - z_2^2 > 0</math>, the inequality<cmath>\frac{8}{(x_1 + x_2)(y_1 + y_2) - (z_1 + z_2)^2} \leq \frac{1}{x_1y_1 - z_1^2} + \frac{1}{x_2y_2 - z_2^2}</cmath>is satisfied. Give necessary and sufficient conditions for equality. |
==Solution== | ==Solution== | ||
Revision as of 20:21, 14 September 2023
Problem
Prove that for all real numbers 
, with 
, the inequality![\[\frac{8}{(x_1 + x_2)(y_1 + y_2) - (z_1 + z_2)^2} \leq \frac{1}{x_1y_1 - z_1^2} + \frac{1}{x_2y_2 - z_2^2}\]](http://latex.artofproblemsolving.com/d/0/6/d064cd5e072e62dbbd229299796392c38859cd85.png)
is satisfied. Give necessary and sufficient conditions for equality.
Solution
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See Also
| 1969 IMO (Problems) • Resources | ||
| Preceded by Problem 5 |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Last Question |
| All IMO Problems and Solutions | ||
