287. Given a trihedral angle, all plane angles of which are right angles, $O$ - the vertex of this angle. Consider all possible broken lines of length $a$, starting at point $O$ and such that any plane parallel to one of the faces of the angle intersects this broken line in no more than one point. Find the geometric locus of the ends of this broken line.
287. Let the edges of a trihedral angle be the coordinate axes. Let $(x, y, z)$ be the coordinates of the vector $\bar{O} \vec{A},\left(x_{i}, y_{i}, z_{i}\right)$ be the coordinates of the $i$-th side of the broken line. Each side of the broken line is considered as a vector. Then

$$
x=\Sigma x_{i}, y=\Sigma y_{i}, z=\Sigma z_{i}
$$

From the condition of the problem, it follows that all $x_{i}$ are non-zero and $\mathbf{c}_{\text { coincide in direction with } x}$ (the same is true for $y_{i}$ and $z_{i}$). Obviously, $O A \mid \leqslant a$. On the other hand,

$$
|x|+|y|+|z|=\sum\left(\left|x_{i}\right|+\left|y_{i}\right|+\left|z_{i}\right|\right) \geqslant \Sigma l_{i}=a
$$

$\left(l_{i}\right.$ - the length of the $i$-th segment of the broken line).

It is not difficult to show that any point $A$, satisfying the conditions $|O A| \leqslant a,|x|+|y|+|z|>a$, where $x, y, z-$ are the coordinates of the point $A$, can be the end of a broken line having no more than three segments, satisfying the conditions of the problem. Let, for example, $M_{1}$
and $M_{2}$ be two points located on the same straight line emanating from the point $O$ such that $\left|x_{1}\right|+\left|y_{1}\right|+\left|z_{1}\right|=a, x_{1} y_{1} z_{1} \neq 0\left(x_{1}, y_{1}\right.$, $z_{1}$ - the coordinates of the point $\left.M_{1}\right),\left|O M_{2}\right|=a$. Consider a broken line with vertices $(0,0,0),\left(x_{1}, 0,0\right),\left(x_{i}, y_{1}, 0\right),\left(x_{1}, y_{1}, z_{1}\right)$. The length of this broken line is $a$. "Stretching" this broken line, we obtain all points of the segment $M_{1} M_{2}$ (excluding $M_{1}$). Thus, the desired geometric locus of points consists of all points outside the octahedron $|x|+|y|+$ $+|z|=a$ and inside or on the surface of the sphere with center at $O$ and radius $a$. At the same time, points located in the coordinate planes are excluded.