## Task A-1.2.

For seven consecutive days, Izabela solved one math test each day. On each test, she achieved a different number of points - the lowest being 91 and the highest being 100. After each test, the average of her results so far was a natural number, and on the seventh test, she scored 95 points.

How many points did Izabela score in total on all seven tests? How many points did she score on the sixth test?
## First Solution.

The average score that Izabela achieved over all seven days is an integer, so we know there exists some natural number $k$ such that Izabela's total score is equal to $7k$.

Furthermore, all the scores are distinct, so the total score is at least

$$
91+92+93+94+95+96+97=658=7 \cdot 94
$$

and at most

$$
94+95+96+97+98+99+100=679=7 \cdot 97
$$

Thus, $k$ is equal to one of the numbers 94, 95, 96, 97.

Furthermore, Izabela scored $7k - 95$ points in the first six days, and this number is divisible by 6. The only possibility is $k=95$.

Therefore, Izabela scored a total of $7 \cdot 95 = 665$ points on all seven tests.

Let $x$ be the number of points Izabela scored on the sixth day. Then she scored $6 \cdot 95 - x$ points in the first five days. This number must be divisible by 5, and since the number 95 is divisible by 5, we conclude that the number $x$ must also be divisible by 5.

Among the numbers 91, 92, 93, 94, 95, 96, 97, 98, 99, and 100, only 95 and 100 are divisible by 5. Since Izabela scored 95 points on the seventh day, and all the results are distinct, we conclude that she scored 100 points on the sixth day.