## Task 1 - 140911

Given a cube with vertices $A, B, C, D, A^{\prime}, B^{\prime}, C^{\prime}, D^{\prime}$ and edge length $a$ (see figure). On $B B^{\prime}$ lies a point $X$, on $B^{\prime} C^{\prime}$ a point $Y$, and on $A^{\prime} B^{\prime}$ a point $Z$, where these points can be arbitrarily placed but must be different from $B^{\prime}$.

We then consider for each such choice of $X, Y, Z$ the closed polygonal chain $X Y Z X$. The length of this polygonal chain is the sum of the lengths $\overline{X Y}, \overline{Y Z}$, and $\overline{Z X}$.

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-0968.jpg?height=339&width=376&top_left_y=710&top_left_x=840)

a) Determine whether there is a polygonal chain with the greatest length among these!

b) Determine whether there is a polygonal chain with the smallest length among these!

c) If there is such a polygonal chain in a) or b), determine its length!
I. For all points $X, Y, Z$ that meet the requirements of the problem, if we set $\left|B^{\prime} X\right|=x,\left|B^{\prime} Y\right|=y$, $\left|B^{\prime} Z\right|=z$, then $0<x, y, z \leq a$. Furthermore, according to the Pythagorean theorem,

$$
|X Y|=\sqrt{x^{2}+y^{2}}, \quad|Y Z|=\sqrt{y^{2}+z^{2}}, \quad|Z X|=\sqrt{x^{2}+z^{2}}
$$

From this it follows that

$$
0<|X Y|+|Y Z|+|Z X| \leq 3 \sqrt{2 a^{2}}
$$

II. For the permissible choice according to the problem statement $X=B, Y=C^{\prime}, Z=A^{\prime}$, we get $x=y=z=a$, thus

$$
|X Y|+|Y Z|+|Z X|=3 \sqrt{2 a^{2}}
$$

III. If, for any points $X, Y, Z$ that meet the requirements of the problem, we choose a point $X *$ between $B^{\prime}$ and $X$ that also meets the requirements of the problem and set $\left|B^{\prime} X *\right|=x *$, then $0<x *<x$, thus

$$
|X * Y|+|Y Z|+|Z X *|=\sqrt{x *^{2}+y^{2}}+\sqrt{y^{2}+z^{2}}+\sqrt{z^{2}+x *^{2}}<|X Y|+|Y Z|+|Z X|
$$

From I. and II. it follows:

a) There is a path among those mentioned in the problem that has the greatest length, namely the one that results for $X=B, Y=C^{\prime}, Z=A^{\prime}$.

c) Its length is $3 a \sqrt{2}$.

## From III it follows:

b) For every path mentioned, there is one with a smaller length, i.e., there is no path with the smallest length.