1978 USAMO Problems/Problem 1

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

1978 USAMO (Problems • Resources)
Preceded by First Question	Followed by Problem 2
1 • 2 • 3 • 4 • 5
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1978 USAMO Problems/Problem 1

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