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Real numbers a, b, c, l, m, n satisfy a*l + b*m + c*n = 0, b*l + c*m + a*n = 0 and c*l + a*m + b*n = 0. If a, b, c are distinct and f(x) = a*x³ + b*x² + c*x + 5, find the value of f(1).

1. 5
2. 0
3. 8
4. 3

Correct answer: 5

Solution

For (l, m, n) not all zero the 3x3 coefficient determinant must be 0. That determinant equals a³ + b³ + c³ - 3abc = (a + b + c)(a² + b² + c² - ab - bc - ca). Since a, b, c are distinct, the quadratic factor a² + b² + c² - ab - bc - ca = (1/2)[(a-b)² + (b-c)² + (c-a)²] is strictly positive, so a + b + c = 0. Then f(1) = a + b + c + 5 = 0 + 5 = 5.

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