5. From the natural numbers from 1 to 99, 50 different numbers were chosen such that no two of them add up to 100 and no two of them add up to 99. Which numbers were chosen?
Answer: numbers from 50 to 99.

Solution. Let's divide the numbers into 49 pairs $(1,99),(2,98), \ldots,(49,51)$, and the number 50 will remain unpaired. In each pair, the sum of the numbers is 100, which means no more than 1 number is chosen from each pair. There are 49 pairs, and a total of 50 numbers are chosen, so exactly 1 number is chosen from each pair, and the remaining number 50 is also chosen. Since 50 is chosen, 49 cannot be taken, and from the pair $(49,51)$, we need to take 51. But since we took 51, we cannot take 48. Continuing this reasoning, we get that we need to take the larger number from each pair.

Criteria. Only the answer - 1 point. The idea of dividing the numbers into 49 such pairs, where the sum of the numbers in them is 100 (or 49 such pairs, where the sum of the numbers in them is 99) - 2 points.