If A, B, C are acute angles such that tanA + tanB + tanC= tanA tanB tanC. Then cotA cotB cotC is

1) ≤ 1 / √3

2) ≤ 1 / 2 √3

3) ≤ 1 / 3 √3

4) none of these

Solution: (3) ≤ 1 / 3 √3

Since AM of 3 quantities ≥ GM of 3 quantities

[tan A + tan B + tan C] / 3 ≥ (tan A tan B tan C)

[tan A + tan B + tan C]3 ≥ 27 (tan A tan B tan C)

(tan A tan B tan C) ≥ 27 (tan A tan B tan C)

(tan A tan B tan C) ≥ 3

(tan A tan B tan C) ≥ (3)3/2 = √27 = 3√3

1 / (tan A tan B tan C) ≤ 1 / 3√3

cot A cot B cot C ≤ 1 / 3√3

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