Difference between revisions of "1978 USAMO Problems/Problem 1"
Mathlobster (talk | contribs) (→Solution 2) |
(→Solution 1) |
||
Line 9: | Line 9: | ||
== Solution 1== | == Solution 1== | ||
− | + | By Cauchy Schwarz, we can see that <math>(1+1+1+1)(a^2+b^2+c^2+d^2)\geq (a+b+c+d)^2</math> | |
− | thus | + | thus <math>4(16-e^2)\geq (8-e)^2</math> |
− | Finally, <math>e(5e-16) \ | + | Finally, <math>e(5e-16) \geq 0</math> which means <math>\frac{16}{5} \geq e \geq 0</math> |
− | + | so the maximum value of <math>e</math> is <math>\frac{16}{5}</math>. | |
'''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== | == Solution 2== | ||
Seeing as we have an inequality with constraints, we can use Lagrange multipliers to solve this problem. | Seeing as we have an inequality with constraints, we can use Lagrange multipliers to solve this problem. |
Revision as of 18:40, 23 February 2018
Contents
Problem
Given that are real numbers such that
,
.
Determine the maximum value of .
Solution 1
By Cauchy Schwarz, we can see that thus Finally, which means so the maximum value of is .
from: Image from Gon Mathcenter.net
Solution 2
Seeing as we have an inequality with constraints, we can use Lagrange multipliers to solve this problem. We get the following equations:
If , then according to and according to , so . Setting the right sides of and equal yields . Similar steps yield that . Thus, becomes and becomes . Solving the system yields , so the maximum possible value of is .
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.