Question 15
We are given 10 real numbers $a_1, a_2, ..., a_{10}$, and we form the 45 sums of two of these numbers $a_i + a_j$, $1 \le i \lt j \le 10$. It is known that not all these sums are integers. Determine the minimum value of $k$ such that it is possible that among the 45 sums there are $k$ that are not integers and $45 - k$ that are integers.