Can a^3+b^3+c^3 Equal d^3+e^3 Given Specific Conditions?

[anemone]

Problem:

Let $a$, $b$, $c$, $d$ and $e$ be strictly positive real numbers such that:

\(\displaystyle a^2+b^2+c^2=d^2+e^2\)

\(\displaystyle a^4+b^4+c^4=d^4+e^4\)

Compare \(\displaystyle a^3+b^3+c^3\) with \(\displaystyle d^3+e^3\).

I have been exhausting all kinds of algebraic tricks, but I still don't get anywhere near to cracking it, and it is so frustrating not knowing how to solve it...

Any help and/or suggestions toward how to solve this problem is much appreciated.

 

[tkhunny]

Have you considered limiting the Domain?

1) What happens when all of a, b, c, d, e are greater than 1?
2) Everything is 1
3) All are between 0 and 1.

 

[Mathwebster]

First, I see some restrictions on $a, b$ and $c$. If you eliminate say $e$ from the pair of equations you get

$d^4 - (a^2+b^2+c^2)d^2 + a^2b^2 + a^2c^2 + b^2c^2 = 0$

Since $d$ is real this means that

$(a^2+b^2+c^2)^2-4(a^2b^2 + a^2c^2 + b^2c^2) \ge 0$

or

$a^4+b^4+c^4- 2 a^2b^2- 2 a^2c^2 -2 b^2c^2 \ge 0$

Also note that if $a=b=c = 1$ then we have a contradiction so this can't happen.

 

[anemone]

Have you considered limiting the Domain?

1) What happens when all of a, b, c, d, e are greater than 1?
2) Everything is 1
3) All are between 0 and 1.

Thanks for the hint, tkhunny...and I think we can rule out the possibility when all of them are 1 because

\(\displaystyle 1^2+1^2+1^2 \ne 1^2+1^2\) and \(\displaystyle 1^4+1^4+1^4 \ne 1^4+1^4\)

But for the case where all of the variables are between 0 and 1, do you mean to suggest that we could let, for example a=cosA as another way to approach the problem?

First, I see some restrictions on $a, b$ and $c$. If you eliminate say $e$ from the pair of equations you get

$d^4 - (a^2+b^2+c^2)d^2 + a^2b^2 + a^2c^2 + b^2c^2 = 0$

Since $d$ is real this means that

$(a^2+b^2+c^2)^2-4(a^2b^2 + a^2c^2 + b^2c^2) \ge 0$

or

$a^4+b^4+c^4- 2 a^2b^2- 2 a^2c^2 -2 b^2c^2 \ge 0$

Also note that if $a=b=c = 1$ then we have a contradiction so this can't happen.

Thanks Jester for showing me some insight that I would not have thought of it myself...but, what should I do after that? I am so confused...

 

[CellCoree]

I understand the frustration of not being able to solve a problem. However, it is important to approach it with a systematic and logical mindset.

One possible approach to this problem is to use the given information to manipulate the equations. First, we can expand the terms in the first two equations to get:

a^4+2a^2b^2+2a^2c^2+b^4+2b^2c^2+c^4=d^4+2d^2e^2+e^4

Next, we can subtract the second equation from the first to eliminate the higher powers of a, b, and c:

2a^2b^2+2a^2c^2+2b^2c^2=2d^2e^2

We can then divide both sides by 2 to get:

a^2b^2+a^2c^2+b^2c^2=d^2e^2

This equation looks familiar because it is the same as the first given equation. Therefore, we can substitute it in to the second equation to get:

a^4+b^4+c^4=d^4+e^4

This means that a^3+b^3+c^3 and d^3+e^3 are equal, as they are the cubes of the terms in the equation above.

In conclusion, through algebraic manipulation and substitution, we have shown that a^3+b^3+c^3 is equal to d^3+e^3. This means that the two expressions are equal and cannot be compared. Both are simply equal to the sum of the cubes of the terms in the first given equation. I hope this helps in solving the problem.