Prove that the nine lightest weights are bronze, and the nine heaviest are iron.

## Solution

The difference between the total weight of the nine heaviest weights and the total weight of the nine lightest weights is $(19+18+\ldots+11)-(9+8+\ldots+1)=90$ grams. Therefore, the iron weights are the heaviest, and the bronze weights are the lightest (otherwise, the difference between the total weight of the iron weights and the total weight of the bronze weights would be less). Thus, the gold weight weighs 10 grams.
Proizvolov V.V.

There are 19 weights of $1, 2, 3, \ldots, 19$ grams: nine iron, nine bronze, and one gold. It is known that the total weight of all iron weights is 90 grams more than the total weight of the bronze weights. Find the weight of the gold weight.