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In triangle ABC the sides opposite to vertices A, B, C are a, b, c respectively. If a², b², c² are the three roots of the equation x³ - p*x² + q*x - k = 0, which of the following is CORRECT?

1. p = a² + b² + c², q = a²*b² + b²*c² + c²*a², k = a²*b²*c²
2. p = a + b + c, q = a*b + b*c + c*a, k = a*b*c
3. p = a² + b² + c², q = a*b*c, k = a²*b²*c²
4. p = (a + b + c)², q = (a*b + b*c + c*a)², k = (a*b*c)²

Correct answer: p = a² + b² + c², q = a²*b² + b²*c² + c²*a², k = a²*b²*c²

Solution

Vieta's formulas for x³ - p*x² + q*x - k = 0 with roots a², b², c² give: sum = a² + b² + c² = p; sum of pairwise products = a²*b² + b²*c² + c²*a² = q; product = a²*b²*c² = k. Note that k = (a*b*c)².

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