Problem 14 · 2020 Math Kangaroo
Stretch
Algebra & Patterns
substitution
Let a, b, c be nonzero real numbers such that \((a - a^{-1})^2 + (b - b^{-1})^2 + (c - c^{-1})^2 = 0\). Which of the following can NOT be the value of \(a + b + c\)?
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Answer: C — 0
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Hint 1 of 2
A sum of three squares equals zero only if each square is zero.
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Hint 2 of 2
Solve x − 1/x = 0; what can x be?
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Approach: zero sum of squares forces each term zero
- Since each squared term is ≥ 0 and they sum to 0, each must be 0: a = 1/a, and likewise for b and c.
- So a, b, c each equal +1 or −1.
- Then a+b+c is one of −3, −1, 1, 3 — never 0, so the impossible value is 0.
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