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If A, B, and C are acute positive angles satisfying A + B + C = π, and cot A cot B cot C = K, which of the following is true?

1. K ≤ (1)/(3√(3))
2. K ≥ (1)/(3√(3))
3. K < (1)/(9)
4. K > (1)/(3)

Correct answer: K ≤ (1)/(3√(3))

Solution

The inequality holds because the product of the cotangents of angles that sum to π is maximized under the constraint of the angles being acute, leading to the conclusion that their product cannot exceed rac{1}{3 ext{√}3}.

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