2. Let $a_{1}, a_{2}, a_{3}, a_{4}$, and $a_{5}$ be distinct real numbers. Denote by $m$ the number of distinct numbers of the form $a_{i}+a_{j}$, where $1 \leq i<j \leq 5$. What is the smallest possible value of the number $m$?
IV /2. We arrange the numbers in order: $a_{1}<a_{2}<\ldots<a_{5}$. Then we have

$$
a_{1}+a_{2}<a_{1}+a_{3}<a_{1}+a_{4}<a_{1}+a_{5}<a_{2}+a_{5}<a_{3}+a_{5}<a_{4}+a_{5}
$$

At least 7 numbers of the form $a_{i}+a_{j}$ for $i<j$ are distinct. On the other hand, for $a_{i}=i$ for $i=1,2,3,4,5$, there are only 7 distinct sums among the $a_{i}+a_{j}$. The sought minimum number $m$ is therefore 7.

Assumption $a_{1}<a_{2}<\ldots<a_{5}: 2$ points. Estimation $a_{1}+a_{2}<a_{1}+a_{3}<a_{1}+a_{4}<a_{1}+a_{5}<a_{2}+a_{5}<a_{3}+a_{5}<a_{4}+a_{5}: 3$ points. Proof that for $a_{i}=i, i=1,2,3,4,5$, the equality is achieved: 2 points. If the contestant does not solve the problem in general, but concludes from the example $a_{i}=i$, $i=1,2,3,4,5$, that the sought number $m$ is 7: 2 points.