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In triangle ABC, if b² + c² = 1999 * a², find the value of (cot B + cot C) / cot A.

1. 1/999
2. 1/1999
3. 999
4. 1999

Correct answer: 1/999

Solution

Using the formula cot X = (sum of squares of adjacent sides - opposite side squared) / (4 * area): cot B + cot C = [(a²+c²-b²) + (a²+b²-c²)] / (4*Delta) = 2a² / (4*Delta). And cot A = (b²+c²-a²)/(4*Delta). So (cot B + cot C)/cot A = 2a² / (b²+c²-a²). Substituting b²+c² = 1999*a²: denominator = 1999*a² - a² = 1998*a². Ratio = 2a² / (1998*a²) = 1/999.

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