Step 1: Given Information - \( A, B, C \) are acute positive angles such that \( A + B + C = \pi \). - \( \cot A \cot B \cot C = K \). Step 2: Use Trigonometric Identities For angles \( A, B, C \) such that \( A + B + C = \pi \), the following identity holds: \[ \cot A \cot B + \cot B \cot C + \cot C \cot A = 1 \] Step 3: Apply the AM-GM Inequality The arithmetic mean-geometric mean (AM-GM) inequality states that for non-negative real numbers \( x, y, z \): \[ \frac{x + y + z}{3} \geq \sqrt[3]{xyz} \] Apply this to \( \cot A, \cot B, \cot C \): \[ \frac{\cot A + \cot B + \cot C}{3} \geq \sqrt[3]{\cot A \cot B \cot C} \] Let \( \cot A \cot B \cot C = K \). Then: \[ \frac{\cot A + \cot B + \cot C}{3} \geq \sqrt[3]{K} \] Step 4: Use the Identity to Relate \( \cot A, \cot B, \cot C \) From the identity: \[ \cot A \cot B + \cot B \cot C + \cot C \cot A = 1 \] Using the AM-GM inequality on \( \cot A \cot B, \cot B \cot C, \cot C \cot A \): \[ \frac{\cot A \cot B + \cot B \cot C + \cot C \cot A}{3} \geq \sqrt[3]{(\cot A \cot B)(\cot B \cot C)(\cot C \cot A)} \] Simplify the right-hand side: \[ \frac{1}{3} \geq \sqrt[3]{(\cot A \cot B \cot C)^2} \] Substitute \( \cot A \cot B \cot C = K \): \[ \frac{1}{3} \geq \sqrt[3]{K^2} \] Raise both sides to the power of 3: \[ \left( \frac{1}{3} \right)^3 \geq K^2 \] \[ \frac{1}{27} \geq K^2 \] Take the square root of both sides: \[ K \leq \frac{1}{3\sqrt{3}} \] Step 5: Conclusion The value of \( K \) satisfies: \[ K \leq \frac{1}{3\sqrt{3}} \] Final Answer: The correct option is: \[ \boxed{A} \]