# Task 4. (12 points)

A shooting tournament involves several series of 10 shots each. In one series, Ivan scored 82 points, as a result of which the average number of points he scored per series increased from 75 to 76 points. How many points does Ivan need to score in the next series of shots to make the average number of points scored per series equal to 77?
# Solution.

Let $N$ be the points scored in the $n$ considered series, in the last of which Ivan scored 82 points. Then $N=76n$ and $N-82=75(n-1)$. Solving the obtained system, we find $n=7$ and $N=532$.

Let $x$ be the points Ivan needs to score to meet the condition of the problem. In this case, we get $77 \cdot 8 = 532 + x \Rightarrow x = 84$.

Answer: 84.

| Criterion Content | Evaluation | Points |
| :---: | :---: | :---: |
| Complete solution. | + | 12 |
| All main logical steps of the solution are provided. There is no strict justification for some conclusions. The answer is correct. | $\pm$ | 9 |
| All main logical steps of the solution are justified. The answer is incorrect due to a computational error or typo. |  |  |
| All main logical steps of the solution are provided. The system of equations is correctly set up. The solution of the system contains errors or is not completed. | $+/ 2$ | 6 |
| The main idea of the solution is found. The answer is missing or incorrect. | 干 | 2 |
| The answer is correct, but the solution is missing or does not contain substantial progress. |  |  |

All-Russian School Olympiad "Mission Possible. Your Calling - Financier!"

| The solution does not meet any of the criteria described above. | $-/ 0$ | 0 |
| :---: | :---: | :---: |
| Maximum score |  | 12 |