It is known that a, B, C. belong to R +, and a + B + C = 1, and 1 / A + 1 / B + 1 / C > = 9 is proved

1/a+1/b+1/c
=(a+b+c)/a+(a+b+c)/b+(a+b+c)/c
=1+(b+c)/a+1(a+c)/b+1(a+b)/c
=3 + B / C + C / B + A / C + C / A + A / B + B / a (because B / A + A / b > = 2, C / A + A / C > = 2, C / B + B / C > = 2)
>=3+2+2+2
=9

It is known that a, B, C. belong to R +, (A / B + B / C + C / a) (B / A + C / B + / A / C) ≥ 9
Verification: it is known that a, B, C. belong to R +, (A / B + B / C + C / a) (B / A + C / B + A / C) ≥ 9

(a/b+b/c+c/a)(b/a+c/b+a/c)
=3+(bc/aa+aa/bc)+(bb/ac+ac/bb)+(ab/cc+cc/ab)>=3+2+2+2=9

It is known that a, B, C belong to R, a + B + C = 1, the proof is: 1 / (a + 1) + 1 / (B + 1) + (c + 1) > = 9 / 4

The left side is written as: [(a + 1) + (B + 1) + (c + 1)] * [1 / (a + 1) + 1 / (B + 1) + 1 / (c + 1)] expanded to 3 + (a + 1) / (b + 1) + (B + 1) / (a + 1) + (a + 1) / (c + 1) + (c + 1) / (a + 1) + (c + 1) / (B + 1) + (B + 1) / (c + 1) > = 9 (using the mean inequality test) is extended to (a1 + A2 + a3. + an) * [(1 / (A1) + 1 / (

We know that a, B, C ∈ R +, a + B + C = 1, and prove that 1A + 1b + 1c ≥ 9

It is proved that 1A + 1b + 1C = a + B + Ca + A + B + CB + A + B + CC = 3 + (Ba + AB) + (Ca + AC) + (BC + CB) ∧ Ba + ab ≥ 2, Ca + AC ≥ 2, BC + CB ≥ 2. If and only if a = b = C, take the equal sign, ∧ 1A + 1b + 1c ≥ 9

It is known that a, B, C. belong to R +, and a + B + C = 1, and 1 / A + 1 / B + 1 / C > = 9 is proved