Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus
During AMC testing, the AoPS Wiki is in read-only mode and no edits can be made.

Difference between revisions of "1978 USAMO Problems/Problem 1"

(→‎Solution)
Line 18: Line 18:
 
== See Also ==
 
== See Also ==
 
{{USAMO box|year=1978|before=First Question|num-a=2}}
 
{{USAMO box|year=1978|before=First Question|num-a=2}}
 +
{{MAA Notice}}
  
 
[[Category:Olympiad Algebra Problems]]
 
[[Category:Olympiad Algebra Problems]]

Revision as of 18:06, 3 July 2013

Problem

Given that $a,b,c,d,e$ are real numbers such that

$a+b+c+d+e=8$,

$a^2+b^2+c^2+d^2+e^2=16$.

Determine the maximum value of $e$.

Solution

Accordting to Cauchy-Schwarz Inequalities, we can see $(1+1+1+1)(a^2+b^2+c^2+d^2)\geqslant (a+b+c+d)^2$ thus, $4(16-e^2)\geqslant (8-e)^2$ Finally, $e(5e-16) \geqslant 0$ that mean, $\frac{16}{5} \geqslant e \geqslant 0$ so the maximum value of $e$ is $\frac{16}{5}$

from: Image from Gon 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

These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. AMC Logo.png

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=1978_USAMO_Problems/Problem_1&oldid=53449"