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The tetrahedron $SABC$ has the following property: there exist five spheres, each tangent to the edges $SA,SB,SC,BC,CA,AB$, or to their extensions.

(a) Prove that the tetrahedron $SABC$ is regular.

(b) Prove conversely that for every regular tetrahedron five such spheres exist.

In tetrahedron $ABCD$, vertex $D$ is connected with $D_0$ the centroid of $\triangle ABC$. Lines parallel to $DD_0$ are drawn through $A, B$ and $C$. These lines intersect the planes $BCD, CAD$ and $ABD$ in points $A_1, B_1$ and $C_1$, respectively. Prove that the volume of $ABCD$ is one third the volume of $A_1B_1C_1D_0$. Is the result true if point $D_0$ is selected anywhere within $\triangle ABC$?

Given the tetrahedron $ABCD$ whose edges $AB$ and $CD$ have lengths $a$ and $b$ respectively. The distance between the skew lines $AB$ and $CD$ is $d$, and the angle between them is $\omega$. Tetrahedron $ABCD$ is divided into two solids by plane $\varepsilon$, parallel to lines $AB$ and $CD$. The ratio of the distances of $\varepsilon$ from $AB$ and $CD$ is equal to $k$. Compute the ratio of the volumes of the two solids obtained.

Prove: The sum of the distances of the vertices of a regular tetrahedron from the center of its circumscribed sphere is less than the sum of the distances of these vertices from any other point in space.

Prove that if one and only one edge of a tetrahedron is greater than 1, then its volume is $\leq 1/8$.

Prove that in every tetrahedron there is a vertex such that the three edges meeting there have lengths which are the sides of a triangle.

For each value of $k = 1, 2, 3, 4, 5$, find necessary and sufficient conditions on the number $a > 0$ so that there exists a tetrahedron with $k$ edges of length $a$, and the remaining $6 - k$ edges of length 1.

In the tetrahedron $ABCD$, angle $BDC$ is a right angle. Suppose that the foot $H$ of the perpendicular from $D$ to the plane $ABC$ is the intersection of the altitudes of $\triangle ABC$. Prove that $$(AB + BC + CA)^2 \leq 6(AD^2 + BD^2 + CD^2).$$

For what tetrahedra does equality hold?

All the faces of tetrahedron $ABCD$ are acute-angled triangles. We consider all closed polygonal paths of the form $XYZTX$ defined as follows: $X$ is a point on edge $AB$ distinct from $A$ and $B$; similarly, $Y, Z, T$ are interior points of edges $BC$, $CD$, $DA$, respectively. Prove:

(a) If $\angle DAB + \angle BCD \neq \angle CDA + \angle ABC$, then among the polygonal paths, there is none of minimal length.

(b) If $\angle DAB + \angle BCD = \angle CDA + \angle ABC$, then there are infinitely many shortest polygonal paths, their common length being $2AC\sin(\alpha/2)$, where $\alpha = \angle BAC + \angle CAD + \angle DAB$.

Given four distinct parallel planes, prove that there exists a regular tetrahedron with a vertex on each plane.

Volumen kocke $ABCDA_1B_1C_1D_1$ jednak je $V$. Nađite volumen zajedničkog dijela tetraedara $AB_1CD_1$ i $A_1BC_1D$.

Neka su u tetraedru $ABCD$ površine strana $ABD$, $ACD$, $BCD$ i $BCA$ redom jednake $S_{1}$, $S_{2}$, $Q_{1}$, $Q_{2}$, a prostorni kut između strana $ABD$ i $ACD$ jednak $\alpha$, odnosno $\beta$ između $BCD$ i $BCA$. Dokažite da je $$S_{1}^{2} + S_{2}^{2} - 2S_{1}S_{2}\cos\alpha = Q_{1}^{2} + Q_{2}^{2} - 2Q_{1}Q_{2}\cos\beta.$$

Svi bridni kutovi pri vrhu $D$ tetraedra $ABCD$ jednaki su $\alpha$, a kutovi između dviju strana tetraedra kojima je jedan vrh $D$ jednaki su $\varphi$. Dokažite da postoji točno jedan kut $\alpha$ za koji je $\varphi = 2\alpha$.

Visine trostrane piramide sijeku se u jednoj točki. Dokažite da ta točka, težište jedne strane piramide, nožište visine na tu stranu i tri točke koje dijele preostale tri visine u omjeru $2:1$, počevši od vrha piramide, leže na istoj sferi.

Dan je tetraedar kojem je jedan brid duljine $3$, a svi ostali duljine $2$. Odredi obujam tog tetraedra.

Osnovka trostrane piramide je trokut sa stranicama duljina $a$, $b$ i $c$. Nasuprotni bridovi su duljina $m$, $n$ i $p$. Dokažite da udaljenost vrha piramide od težišta osnovke iznosi $$\frac{1}{3}\sqrt{3(m^2 + n^2 + p^2) - (a^2 + b^2 + c^2)}.$$

Polovištem svakog brida tetraedra položena je ravnina okomito na suprotni brid. Dokažite da se svih šest ravnina siječe u točki koja je simetrična središtu opisane sfere tetraedra u odnosu na njegovo težište.