1. On the board, five different positive numbers are written. Determine the maximum number of ways in which pairs can be formed from them, the sum of which equals one of the five numbers written on the board.
1. Let $a_{1}<a_{2}<a_{3}<a_{4}<a_{5}$ be positive numbers written on the board. The smallest numbers $a_{1}$ and $a_{2}$ clearly cannot be the sum of any two numbers written on the board. The number $a_{3}$ can be obtained as the sum of some pair in at most one way, namely $a_{3}=a_{1}+a_{2}$. The number $a_{4}$ can theoretically be obtained in three ways: $a_{1}+a_{2}$, $a_{2}+a_{3}$, or $a_{1}+a_{3}$, but since $a_{1}+a_{2}<a_{1}+a_{3}<a_{2}+a_{3}$, $a_{4}$ can be the sum of at most one of the pairs $\left\{a_{1}, a_{2}\right\}$, $\left\{a_{1}, a_{3}\right\}$, $\left\{a_{2}, a_{3}\right\}$. Finally, the number $a_{5}$ can be obtained in at most two ways. If $a_{5}$ could be obtained in three ways as the sum of two numbers written on the board, at least one of the numbers $a_{1}, a_{2}, a_{3}, a_{4}$ would be a summand in two different sums, which is not possible.

In total, we have found that the number of suitable pairs never exceeds the sum $0+0+1+1+2$, which equals four.

An example of five numbers written on the board, for which the sums of four (of the pairs formed from them) are also listed on the board, is $a_{i}=i$ for $i \in\{1,2,3,4,5\}$, where $1+2=3$, $1+3=4$, $1+4=5$, and $2+3=5$.

Conclusion. From the given five different positive numbers, at most four pairs with the required property can be formed.

For a complete solution, award 6 points. For obtaining the estimate that there are no more than four suitable pairs, award 4 points. (For the estimate that there are no more than six pairs, because the largest number cannot be in a pair, do not award points.) For an example of five numbers with four suitable pairs, award 2 points, even if the solver fails to prove the upper estimate.