I am thinking of several consecutive natural numbers. If we crossed out the numbers 70, 82, and 103, the arithmetic mean of the numbers would not change. If we instead crossed out the numbers 122 and 123, the arithmetic mean would decrease by exactly 1. Which natural numbers am I thinking of?

(L. Šimůnek)
If by scratching out a triplet of numbers from the set, the arithmetic mean of the set does not change, then this triplet of numbers has the same arithmetic mean as the entire set. The arithmetic mean of the entire set of numbers is thus

$$
\frac{70+82+103}{3}=85
$$

Let the number of numbers in the entire set be $n$. After scratching out the numbers 122 and 123, $n-2$ numbers remain, and according to the problem, the arithmetic mean decreases by 1, i.e., it becomes 84. By multiplying the arithmetic mean of the numbers by the number of these numbers, we get their sum. Based on this, we set up the following equation and solve it:

$$
\begin{aligned}
85 n & =84(n-2)+122+123 \\
85 n & =84 n+77, \\
n & =77 .
\end{aligned}
$$

The unknown set of numbers consists of 77 consecutive numbers.

The middle number is equal to the arithmetic mean, with 38 numbers before and after it, since $77=2 \cdot 38+1$. The smallest number in the set is thus $85-38=47$ and the largest $85+38=123$. The sought numbers are the natural numbers from 47 to 123 inclusive.

Alternative solution. In the same way, we determine that the arithmetic mean of the set of numbers before scratching is 85. The scratched-out numbers 122 and 123 can then be expressed using this mean: $122=85+37$, $123=85+38$. In the sum of the numbers that remain after scratching out the numbers 122 and 123, there is exactly the sum $37+38$ missing for these numbers to have an arithmetic mean of 85. From the problem, we know that the arithmetic mean of these remaining numbers is 1 less than 85, so there must be exactly $37+38$, i.e., 75 of these numbers. The number of numbers before scratching was $75+2$, i.e., 77. We proceed as in the previous solution.

Scoring suggestion. 2 points for the mean of the thought numbers (85); 3 points for the number of thought numbers (77); 1 point for the thought numbers (47 to 123).