Problem 21 · 2014 Math Kangaroo
Hard
Algebra & Patterns
last-digitcasework
Let \(a,b,c\) be different real numbers, none equal to zero, and let \(n\) be a positive whole number. It is known that the numbers \((-2)^{2n+3} imes a^{2n+2} imes b^{2n-1} imes c^{3n+2}\) and \((-3)^{2n+2} imes a^{4n+1} imes b^{2n+5} imes c^{3n-4}\) have the same sign. Which of the following statements is definitely true?
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Answer: D — \(a<0\)
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Hint 1 of 2
Even powers are always positive, so only the odd-powered factors carry a sign.
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Hint 2 of 2
Compare the parities of the exponents in the two products to see what must be true.
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Approach: track signs through even/odd exponents
- In a product, only factors with odd exponents affect the sign; even powers are positive.
- Matching the two products' signs forces the contribution of a to flip consistently, and the only sign that is pinned down in every case is that of a.
- Working it through shows a must be negative, so (D): a < 0 is definitely true.
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