1. Given are fifty natural numbers, of which half do not exceed 50, and the other half are greater than 50 but less than 100. The difference between any two of the given numbers is not 0 or 50. Find the sum of these numbers.
Solution. If we subtract 50 from the given natural numbers, which are greater than 50, none of the obtained differences will be equal to any of the remaining numbers, which follows that the difference between any two of the given numbers is not 50. Then the 25 differences and the 25 given numbers, less than 50, are different natural numbers from 1 to 50, i.e., all natural numbers from 1 to 50. Their sum is

$$
1+2+\ldots+50=\frac{50.51}{2}=25 \cdot 51=1275
$$

so the sum of all the given numbers is $1275+25 \cdot 50=1275+1250=2525$.