Cho a,b,c > 0 t/m c^2 = a^3 + b^3 = 3ab + c

To analyze the conditions given in the problem, let's first rewrite the relations based on the assumptions:

1. \( c^2 = a^3 + b^3 \)
2. \( c^2 = 3ab + c \)

From the first equation, we can express \( c \) as:
\[
c = \sqrt{a^3 + b^3}
\]

Substituting \( c \) into the second equation gives:
\[
a^3 + b^3 = 3ab + \sqrt{a^3 + b^3}
\]

To solve this, we set \( x = c^2 \), leading to:
\[
x = a^3 + b^3
\]
And from the second equation:
\[
x = 3ab + \sqrt{x}
\]

This gives us a non-linear equation in terms of \( x \):
\[
x - \sqrt{x} - 3ab = 0
\]

Let \( t = \sqrt{x} \), giving:
\[
t^2 - t - 3ab = 0
\]

We can solve this for \( t \) using the quadratic formula:
\[
t = \frac{1 \pm \sqrt{1 + 12ab}}{2}
\]
Since \( t \) (which is \( \sqrt{x} \)) must be positive, we take the positive root:
\[
t = \frac{1 + \sqrt{1 + 12ab}}{2}
\]

Now squaring \( t \), we find \( c^2 = x = t^2 \):
\[
c^2 = \left(\frac{1 + \sqrt{1 + 12ab}}{2}\right)^2 = \frac{(1 + \sqrt{1 + 12ab})^2}{4}
\]
Calculating this results in:
\[
c^2 = \frac{1 + 2\sqrt{1 + 12ab} + (1 + 12ab)}{4} = \frac{2 + 12ab + 2\sqrt{1 + 12ab}}{4}
\]

At this step, we've derived an expression for \( c^2 \) in terms of \( a \) and \( b \). Given the challenge to prove or derive other conditions like \( 2a^2b^2 + 2b^2c^2 + 2c^2a^2 = a^4 + b^4 + c^4 \), we can apply algebraic manipulations or symmetry properties among \( a, b, c \).

From the initial conditions used, we can derive more expressions or utilize relationships such as the AM-GM inequality. You may employ numerical values or specific scenarios for \( a, b, c \) to explore scenarios in proving symmetric properties.

The main approach revolves around inter-relating these variables and checking for identities or symmetries between them, which can lead to proving the equality you've presented.

Due to the complexity of \( CMR : 2.a^2b^2 + 2.b^2c^2 + 2.c^2a^2 = a^4 + b^4 + c^4 \), you may wish to expand both sides and try comparing coefficients or using specific values for \( a, b, \) and \( c \) to verify. An algebraic proof may utilize variable substitutions or inequalities, depending on how you proceed with simplifications from known relations.