Problem 12 · 2022 Math Kangaroo
Hard
Algebra & Patterns
substitution
a, b and c are real numbers not equal to zero. It is known that the numbers \(-2a^4b^3c^2\) and \(3a^3b^5c^{-4}\) have the same sign. Which of the following statements is definitely correct?
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Answer: E — \(a<0\)
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Hint 1 of 2
Track only the signs: even powers are positive, so focus on the odd-power factors and the leading constants.
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Hint 2 of 2
Set the two signs equal and solve for the sign of a.
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Approach: sign analysis
- Sign of −2a^4 b^3 c^2 is −sign(b) (the a^4 and c^2 are positive).
- Sign of 3a^3 b^5 c^(−4) is sign(a)·sign(b) (c^(−4) is positive).
- Equal signs: −sign(b) = sign(a)·sign(b), so sign(a) = −1.
- Therefore a < 0.
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