[ Cauchy's Inequality ]

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A seller has a balance with unequal arms and weights. First, he weighs the goods on one pan, then on the other, and takes the average weight. Is he cheating?

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Let's say we put a commodity weighing 1 kg on the left pan of the scales, and on the other pan for balance - weights weighing $a$ kg, that is, $a$ times more. If we move the commodity to the right pan, then for balance on the left, we will need to put a weight $a$ times less, that is, $1 / a$ kg. Thus, the seller will take money for $1 / 2(a + 1 / a)$ kg, which, according to the Cauchy inequality, is more than 1 kg.

## Answer

He is cheating.