Tokarev S.I.

Given the numbers $1,2, \ldots, N$, each of which is colored either black or white. It is allowed to repaint in the opposite color any three numbers, one of which is equal to the half-sum of the other two. For which $N$ is it always possible to make all numbers white?
If $k$ is the largest black number, then we can repaint the triplet of numbers $(k-2, k-1, k)$ and proceed similarly until all numbers, except possibly 1 and 2, become white.

If the number 2 is still black after this, then for $N \geq 8$, by repainting the triplets $(2,5,8),(5,6,7)$, and $(6,7,8)$, it can be made white (the color of the other numbers will not change); then, if necessary, the color of the number 1 can also be changed (by repainting the triplets $(1,4,7),(4,5,6)$, and $(5,6,7)$).

The cases $N=1$ and $N=2$ are obvious, and for the cases $3 \leq N \leq 7$, note that the parity of the number of black numbers in the set $\{2,3,5,6\}$ does not change during the repaintings.

Therefore, if initially the number 2 is black and the others are white, it is impossible to make all numbers white.

## Answer

For $N \geq 8$.