The number of real solutions for x for the equation $(a^2)^x+(b^2)^x+(c^2)^x=(ab)^x+(bc)^x+(ca)^x$ if a, b, c are unequal real numbers:

(1) 3 solutions
(2) 1 solution
(3) No solution
(4) Infinite solutions

Solution

We have, $a^{2x}+b^{2x}+c^{2x}=a^x b^x + b^x c^x + c^x a^x $

Or, $(a^x)^2+(b^x)^2+(c^x)^2=a^x b^x + b^x c^x + c^x a^x $

Let, $a^x = X, b^x = Y, c^x = Z$

So, $X^2+Y^2+Z^2=XY+YZ+ZX$

$\Rightarrow (X-Y)^2 + (Y-Z)^2 + (Z-X)^2 =0$

$\therefore X=Y=Z$ Or $a^x = b^x = c^x $

The above is only possible when x = 0 as $a \neq b \neq c $

Hence, option (2)