2. Half-year grade. By the end of the half-year, Vasya Petrov had the following grades in mathematics in his journal: $4,1,2,5,2$. Before assigning the half-year grade, the math teacher told Vasya:

- Vasya, you can choose the method to determine your half-year grade. I offer two options. Method A: the arithmetic mean of the current grades, rounded to the nearest whole number. Method B: the median of the current grades.

The best method for Vasya is the one that gives him the highest grade for the half-year.

![](https://cdn.mathpix.com/cropped/2024_05_06_853fe0fb72821c0a3a48g-10.jpg?height=328&width=534&top_left_y=1912&top_left_x=1366)

a) (from 6th grade. 1 point). Which method is the best for Vasya?

b) (from 6th grade. 2 points). Then the teacher thought and added:

- Keep in mind, Vasya, that if you manage to choose the best method for yourself, I will add two more fives to your journal, and then calculate the half-year grade.

Prove that under these conditions, Method A is not the best for Vasya.
# Solution.

a) The average of the current grades is 2.8 (rounded to 3), and the median of the grades is 2. It is better to choose Method A.

b) Suppose Method A is the best. Therefore, the teacher should give two fives. However, the average after rounding remains 3, while the median rises to 4. Thus, Method A now gives less than Method B, so Method A is not the best.