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Question

# If the product of three positive real numbers is 1 and their sum is greater than sum of their reciprocals, then

A

exactly one of them exceeds 1

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B

each of them equals 1

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C

exactly one of them is less than 1

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D

None of these

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Solution

## The correct option is A exactly one of them exceeds 1 Let the numbers be a, b, 1ab. We are given a+b+1ab>1a+1b+abNow,(a−1)(b−1)(1ab−1)=1+(a+b+1ab)−(ab+1a+1b)−1>0 ⇒ either all three of a-1, b-1, 1ab−1 are positive or exactly one of them is positive. But all three of a-1, b-1, 1ab−1 cannot be positive.

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