1. Heating season. The municipal government of Zatonsk has adopted a rule: heating in houses should be turned on no earlier than October 26, but only if the average temperature over the previous three days is below $8^{\circ} \mathrm{C}$. The city has two districts: Coastal and Riverside.

In the Coastal district, the rule was understood as follows: if the average daily temperature is below $8^{\circ} \mathrm{C}$ for three consecutive days, then on the fourth day, heating should be turned on if this day is October 26 or later.

In the Riverside district, the rule was understood differently: if the average temperature over a three-day period is below $8^{\circ} \mathrm{C}$, then on the fourth day, heating should be turned on if this day is not earlier than October 26.

The table shows the average daily temperature for several days in October.

| Date | Oct 22 | Oct 23 | Oct 24 | Oct 25 | Oct 26 | Oct 27 | Oct 28 | Oct 29 | Oct 30 |
| :--- | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| Avg. Temp. ${ }^{\circ} \mathrm{C}$ | 3 | 9 | 10 | 6 | 3 | 7 | 10 | 6 | 8 |

a) (from 6th grade. 1 point). On what date was heating turned on in the Coastal district? On what date was heating turned on in the Riverside district?

b) (from 6th grade. 1 point). Prove that no matter what days occur in October, heating in the Riverside district will be turned on no later than in the Coastal district.
# Solution.

a) In Zarechny District, the average temperature over a three-day period is calculated. For the period of October 23-25, the average temperature is $8.33^{\circ} \mathrm{C}$, so heating should not be turned on on October 26. However, for the period from October 24 to 26, the average temperature is below $8^{\circ} \mathrm{C}$, which means heating should be turned on on October 27. In Pribrezhny District, they do not calculate the average temperature over three days but instead look at the temperature for each individual day. The first time the average daily temperature was below $8^{\circ} \mathrm{C}$ for three consecutive days was on October 25, 26, and 27. Therefore, heating will be provided in Pribrezhny District only on October 28.

b) If the average daily temperature is below the threshold of $8^{\circ} \mathrm{C}$ for three consecutive days, then the average over all three days will also be below this threshold. The reverse is not true, so heating in Pribrezhny District will not be turned on earlier than in Zarechny District, and possibly later.

Answer: a) in Zarechny District on October 27, in Pribrezhny District on October 28.