The proof of inequality Let a, B, C be positive real numbers and ABC = 1 1/(1+2a)+1/(1+2b)+1/(1+2c)>=1

The proof of inequality Let a, B, C be positive real numbers and ABC = 1 1/(1+2a)+1/(1+2b)+1/(1+2c)>=1

Let me try
General idea: using the method of local inequality,
We construct
1/(1+2a)≥ (a^k)/(a^k+b^k+c^k),
The above formula is equivalent to B ^ k + C ^ k ≥ 2A ^ (K + 1)
This is determined by the mean inequality and ABC = 1
B ^ k + C ^ k ≥ 2 √ (b ^ KC ^ k) = 2 √ (a ^ - K) order = 2A ^ (K + 1)
The solution is k = - 2 / 3
Similarly,
1/(1+2b)≥ (b^k)/(a^k+b^k+c^k),
1/(1+2c)≥ (c^k)/(a^k+b^k+c^k),
Add the three formulas above