6. (20 points) Dasha added 158 numbers and got 1580. Then Seryozha tripled the largest of these numbers and decreased another number by 20. The resulting sum did not change. Find the smallest of the original numbers.

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# Solution:

Let x be the largest of the original numbers, and y be the number that Seryozha decreased. Then: $x+y=3 x+y-20$, i.e., $x=10$.

Since the arithmetic mean of the original numbers is 10, and the largest of these numbers is also 10, each of the given numbers is 10.

Answer: 10.

## Criteria:

| 20 points | Any complete and correct solution. |
| :--- | :--- |
| 15 points | A generally correct solution with minor flaws. |
| 5 points | The largest number is found. |
| 0 points. | The solution does not meet any of the criteria listed above or only the answer is provided. |