# Task No. 8.1

## Condition:

For the "Handy Hands" club, Anton needs to cut several identical pieces of wire (the length of each piece is a whole number of centimeters). Initially, Anton took a piece of wire 10 meters long and was able to cut only 9 needed pieces from it. Then Anton took a piece 11 meters long, but it also only provided 9 pieces. What length of pieces did Anton need to cut? Express the answer in centimeters.
Answer: 111

Exact match of the answer -1 point

## Solution.

First, note that 9 pieces of 111 cm each make up 999 cm. Therefore, both the first and the second piece are enough for 9 parts, but the second piece is not enough for 10 parts: $10 \times 111=1110>1100$.

We will prove that if the length of the piece is not 111 cm, then one of the conditions of the problem will not be met. If the length of the piece is no more than 110 cm, then from a piece of 11 m in length, it would be possible to cut at least 10 parts, which contradicts the condition.

If the length of the piece is at least 112 cm, then 9 parts would require at least 1008 cm, that is, they could not be cut from a ten-meter piece.