No Ordinary Optimization Problem

Calculus Level 3

If \(a, b, c>0\) and \(a+b+c=1\), the minimum value of \((a+\frac{1}{a})^{10}+(b+\frac{1}{b})^{10}+(c+\frac{1}{c})^{10}\) can be expressed as \(\frac{10^{d}}{3^{e}}\) where \(d\) and \(e\) are positive integers. Find \(d+e\)