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Given a, b, c are positive reals, the 3x3 determinant with rows (a/b + a/c, b/c + b/a, 1), (b/c + b/a, c/a + c/b, 1), (a/c + c/a, a/b + a/c, 1) equals 0. Which statement must necessarily hold?

1. 1/a + 1/b + 1/c = 0
2. a² + b² + c² = ab + bc + ac
3. a = b = c
4. 1/a + 1/b + 1/c = 1

Correct answer: a = b = c

Solution

The determinant condition reduces to a² + b² + c² = ab + bc + ca, which for positive reals forces a = b = c.

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