Any neat proof that $0$ is the unique solution of the equation $4^x+9^x+25^x=6^x+10^x+15^x$?

HINT: $$a^2+b^2+c^2\geq ab+bc+ca$$


Since the question is already settled for the continuous version, I thought I would just add (for fun!) a short proof for the case in which $x$ is a whole number: for $x \geq 1$, the two sides are not equal when reduced modulo ten. So, the only possible whole number solution is at $x=0$, which works. "QED"

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Real Analysis

Exponential Function

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