Sum of a Triple

Algebra Level 4

$$a, b$$ and $$c$$ are positive real numbers greater than or equal to 1 satisfying

\begin{align} abc & =100,\\ a^{\lg a} b^{\lg b} c^{\lg c} & \geq 10000. \\ \end{align}

What is the value of $$a + b + c$$?

Details and assumptions:

• $$\lg$$ refers to log base 10.
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