10. (CZS 4) The square $A B C D$ is to be decomposed into $n$ triangles (nonoverlapping) all of whose angles are acute. Find the smallest integer $n$ for which there exists a solution to this problem and construct at
least one decomposition for this n. Answer whether it is possible to ask additionally that (at least) one of these triangles has a perimeter less than an arbitrarily given positive number.
10. Let $n$ be the number of triangles and let $b$ and $i$ be the numbers of vertices on the boundary and in the interior of the square, respectively.
Since all the triangles are acute, each of the vertices of the square belongs to at least two triangles. Additionally, every vertex on the boundary belongs to at least three, and every vertex in the interior belongs to at least five triangles. Therefore
$$
3 n \geq 8+3 b+5 i \text {. }
$$

Moreover, the sum of angles at any vertex that lies in the interior, on the boundary, or at a vertex of the square is equal to $2 \pi, \pi, \pi / 2$ respectively: The sum of all angles of the triangles equals $n \pi$, which gives us $n \pi=4 \cdot \pi / 2+b \pi+2 i \pi$, i.e., $n=$ $2+b+2 i$. This relation together with (1) easily yields that $i \geq 2$. Since each of the vertices inside the square belongs to at least five triangles, and at most two contain both, it follows that $n \geq 8$.
It is shown in the figure that the square can be decomposed into eight acute triangles. Obviously one of them can have an arbitrarily small perimeter.