1. Observing the temperatures recorded in Cesenatico in the last months of December and January, Stefano noticed a strange coincidence: on all the days of this period (excluding the first and last day), the minimum temperature was the sum of the minimum temperature of the previous day and the next day.

Knowing that on December 3 the minimum temperature was 5 degrees, and on January 31 it was 2 degrees, determine the minimum temperature on December 25.
## SOLUTION:

The minimum temperature recorded on December 25th was -3 degrees.

To demonstrate this, let $x$ be the minimum temperature recorded on December 1st and $y$ be the minimum temperature recorded on December 2nd. Using the relationship observed by Stefano, the minimum temperature on a given day can be derived from those of the two preceding days: in this way, the minimum temperatures for the first few days of December are as reported in the following table:

| Day | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| Min. Temp. | $x$ | $y$ | $y-x$ | $-x$ | $-y$ | $x-y$ | $x$ | $y$ | $y-x$ | $-x$ |

It is therefore easily seen that the sequence of minimum temperatures repeats every 6 days. Consequently, the minimum temperature on December 3rd was $y-x$, while that on January 31st, which is the 62nd day of the period, coincides with that of the second, i.e., $y$. From the information, it is deduced that $y-x=5$ and $y=2$, from which $x=-3$.

Now, the minimum temperature on December 25th coincides with that of the 1st, since $25-1$ is a multiple of 6, and was therefore -3 degrees.