Difference between revisions of "1978 USAMO Problems/Problem 1"
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Determine the maximum value of <math>e</math>. | Determine the maximum value of <math>e</math>. | ||
| − | == Solution == | + | == Solution 1== |
Accordting to '''Cauchy-Schwarz Inequalities''', we can see <math>(1+1+1+1)(a^2+b^2+c^2+d^2)\geqslant (a+b+c+d)^2</math> | Accordting to '''Cauchy-Schwarz Inequalities''', we can see <math>(1+1+1+1)(a^2+b^2+c^2+d^2)\geqslant (a+b+c+d)^2</math> | ||
thus, <math>4(16-e^2)\geqslant (8-e)^2</math> | thus, <math>4(16-e^2)\geqslant (8-e)^2</math> | ||
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'''from:''' [http://image.ohozaa.com/view2/vUGiXdRQdAPyw036 Image from Gon Mathcenter.net] | '''from:''' [http://image.ohozaa.com/view2/vUGiXdRQdAPyw036 Image from Gon Mathcenter.net] | ||
| + | == Solution 2== | ||
== See Also == | == See Also == | ||
Revision as of 12:35, 30 April 2016
Contents
Problem
Given that 
are real numbers such that
,
.
Determine the maximum value of 
.
Solution 1
Accordting to Cauchy-Schwarz Inequalities, we can see 
thus, 
Finally, 
that mean, 
so the maximum value of is 
from: Image from Gon Mathcenter.net
Solution 2
See Also
| 1978 USAMO (Problems • Resources) | ||
| Preceded by First Question |
Followed by Problem 2 | |
| 1 • 2 • 3 • 4 • 5 | ||
| All USAMO Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
