## Task 5 - 311235

Determine whether it is possible to select fifty different natural numbers from 1 to 100 such that their sum is 2525 and no two of them add up to 101.
Let $M$ be the set of natural numbers from 14 to 28 inclusive, 30, from 33 to 50 inclusive, from 69 to 70 inclusive, 72, and from 88 to 100 inclusive.

Then $M$ has exactly $15+1+18+2+1+13=50$ different elements, none of which add up to 101, since from each pair $(n, 101-n)$ of natural numbers between 1 and 100 inclusive, exactly one number is contained in $M$. Summing all elements of $M$, we obtain the sum $s$ as follows:

$$
s=15 \cdot \frac{14+28}{2}+30+18 \cdot \frac{33+50}{2}+69+70+72+13 \cdot \frac{88+100}{2}=2525
$$

Thus, it is evident that a corresponding selection that satisfies the problem statement can be made.