MHB Finding Maximum Value of $e$ in $a,b,c,d,e \in R$

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The discussion focuses on maximizing the value of e given the constraints a+b+c+d+e=8 and a^2+b^2+c^2+d^2+e^2=16. It is established that when a, b, c, and d are all equal to 2, e equals 0, while setting a, b, c, and d to 1.2 results in e being 3.2. Participants are encouraged to find the maximum value of e and provide a proof for their solution. A reference to MarkFL's post suggests that it contains relevant insights or solutions. The conversation emphasizes the mathematical challenge of determining e's maximum value under the given conditions.
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$a,b,c,d,e \in R$
$a+b+c+d+e=8$
$a^2+b^2+c^2+d^2+e^2=16$
$find :\,\, e_{max}$
 
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Albert said:
$a,b,c,d,e \in R$
$a+b+c+d+e=8$
$a^2+b^2+c^2+d^2+e^2=16$
$find :\,\, e_{max}$

[sp]
When a=b=c=d=2 we get e=0 and when a=b=c=d=1.2 we get e=3.2.
[/sp]
 
$e_{max}=?$
and can you prove it ?
 
My solution:

Because of the cyclic symmetry in the variables, we may let:

$$a=b=c=d$$

And so:

$$4a+e=8\implies a=\frac{8-e}{4}$$

$$4a^2+e^2=16$$

Substitute for $a$:

$$4\left(\frac{8-e}{4} \right)^2+e^2=16$$

This simplifies to:

$$e(5e-16)=0$$

Hence:

$$e_{\max}=\frac{16}{5}$$
 
Albert said:
$e_{max}=?$
and can you prove it ?

Yep. See MarkFL's post. :p
 

Thread 'Area of Overlapping Squares'

Here is a little puzzle from the book 100 Geometric Games by Pierre Berloquin. The side of a small square is one meter long and the side of a larger square one and a half meters long. One vertex of the large square is at the center of the small square. The side of the large square cuts two sides of the small square into one- third parts and two-thirds parts. What is the area where the squares overlap?
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