Difference between revisions of "1978 USAMO Problems/Problem 1"
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== Solution == | == Solution == | ||
| − | {{ | + | 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> | ||
| + | Finally, <math>e(5e-16) \geqslant 0</math> that mean, <math>\frac{16}{5} \geqslant e \geqslant 0</math> | ||
| + | '''so''' the maximum value of <math>e</math> is <math>\frac{16}{5}</math> | ||
| + | |||
| + | '''from:''' [http://www.mathcenter.net/forum Mathcenter.net] | ||
== See Also == | == See Also == | ||
Revision as of 11:30, 20 March 2013
Problem
Given that 
are real numbers such that
,
.
Determine the maximum value of 
.
Solution
Accordting to Cauchy-Schwarz Inequalities, we can see 
thus, 
Finally, 
that mean, 
so the maximum value of is 
from: Mathcenter.net
See Also
| 1978 USAMO (Problems • Resources) | ||
| Preceded by First Question |
Followed by Problem 2 | |
| 1 • 2 • 3 • 4 • 5 | ||
| All USAMO Problems and Solutions | ||
