## 1. task

At exactly 9 o'clock, Petar lit two candles simultaneously, one 29 cm long and the other 34 cm long. It is known that 2 mm of the shorter candle burns in 4 minutes, and 2 mm of the longer candle in 3 minutes. After some time, Petar noticed that one candle was 2 cm longer than the other. At what time could this have happened?
First method

The shorter candle, after 4 minutes, decreases by $2 \mathrm{~mm}$, so after 1 hour it decreases by $15 \cdot 2=30 \mathrm{~mm}=3 \mathrm{~cm}$.

The longer candle, after 3 minutes, decreases by $2 \mathrm{~mm}$, so after 1 hour it decreases by $20 \cdot 2=40 \mathrm{~mm}=4 \mathrm{~cm}$.

Therefore, the difference in their lengths decreases by $1 \mathrm{~cm}$ every hour.

Since the initial difference in length is $34-29=5 \mathrm{~cm}$, it will take 3 hours for the difference to be $2 \mathrm{~cm}$. Thus, at 12:00.

However, the longer candle burns faster, so let's consider the situation when a smaller part of it remains compared to the shorter candle.

Since the initial difference in length is $5 \mathrm{~cm}$, and the difference decreases by $1 \mathrm{~cm}$ every hour, after 5 hours the candles will be of equal height.

This means that after another 2 hours, the difference in their lengths will again be $2 \mathrm{~cm}$.

Since $29 \div 3 = 9$ with a remainder of 2, and $34 \div 4 = 8$ with a remainder of 2, both candles will burn for more than 8 hours.

This means that the difference will again be $2 \mathrm{~cm}$, 7 hours after the start of burning. Thus, at 16:00.