C

1103233602

Level: 
C
Let $ABCDEFGH$ be a cube with an edge length of $1$ placed in the rectangular coordinate system. In the cube a regular tetrahedron $ACHF$ is highlighted (see the picture). Find the distance between the opposite edges of this tetrahedron.\[ \] Hint: A tetrahedron’s opposite edges lie on skew lines. Their distance is the same as the distance of the midpoint of one edge from the opposite edge.
$1$
$\sqrt3$
$\frac{\sqrt3}2$
$\frac{\sqrt5}2$

1103233601

Level: 
C
Let $ABCDEFGH$ be a cube with an edge length of $1$ placed in the rectangular coordinate system. In the cube a regular tetrahedron $ACHF$ is highlighted (see the picture). Find its perpendicular height. \[ \] Hint: Find e.g. the distance between the point $F$ and the plane $ACH$.
$\frac{2\sqrt3}3$
$\frac{\sqrt3}3$
$\frac{2\sqrt6}3$
$\frac23$

1103040208

Level: 
C
We are given the points $A = [4;5;-1]$, $B = [-2;-1;2]$, $C = [-1;-3;0]$ and $D = [0;m;2]$. Find the missing coordinate of the point $D$ such that the point $D$ lies in the plane determined by the points $A$, $B$ and $C$. Hint: Use a linear combination of vectors shown in the picture or use their mixed product.
$m=3$
$m=-3$
$m=1$
$m$ does not exist

1003040207

Level: 
C
Given the points $A = [2;0;3]$ and $B = [-1;2;0]$, specify all the points $C$ lying on the $z$-axis, such that the area of the triangle $ABC$ is $2\sqrt2$. Hint: Use a cross product of vectors.
$C_1=[0;0;1];\ C_2=\left[0;0;\frac{29}{13}\right]$
$C_1=[0;0;1];\ C_2=\left[0;0;-1\right]$
$C_1=[0;0;-1];\ C_2=\left[0;0;\frac{13}{29}\right]$
$C_1=[0;0;-1];\ C_2=\left[0;0;\frac{29}{13}\right]$