RU-ALLRUSSIAN 2015 Problem 98

$N\geq9$ distinct real numbers are written on a blackboard. All these numbers are nonnegative, and all are less than $1$ . It happens that for very $8$ distinct numbers on the board, the board contains the ninth number distinct from eight such that the sum of all these nine numbers is integer. Find all values $N$ for which this is possible. (F. Nilov)