It is known in Euclidean geometry that the sum of the angles of a triangle is constant. Prove, however, that the sum of the dihedral angles of a tetrahedron is not constant.

A regular triangular pyramid $ABCD$ with the base side $AB=a$ and the lateral edge $AD=b$ is given. Let $M$ and $N$ be the midpoints of $AB$ and $CD$ respectively. A line $\alpha$ through $MN$ intersects the edges $AD$ and $BC$ at $P$ and $Q$, respectively. (a) Prove that $AP/AD=BQ/BC$. (b) Find the ratio $AP/AD$ which minimizes the area of $MQNP$.

(a) Prove the parallelogram law that says that in a parallelogram the sum of the squares of the lengths of the four sides equals the sum of the squares of the lengths of the two diagonals. (b) The edges of a tetrahedron have lengths $a, b, c, d, e$ and $f$. The three line segments connecting the centers of intersecting edges have lengths $x, y$ and $z$. Prove that $$4 (x^2 + y^2 + z^2) = a^2 + b^2 + c^2 + d^2 + e^2 + f^2$$

Let $M$ be a point on the edge $CD$ of a tetrahedron $ABCD$ such that the tetrahedra $ABCM$ and $ABDM$ have the same total areas. We denote by $\pi_{AB}$ the plane $ABM$. Planes $\pi_{AC},...,\pi_{CD}$ are analogously defined. Prove that the six planes $\pi_{AB},...,\pi_{CD}$ are concurrent in a certain point $N$, and show that $N$ is symmetric to the incenter $I$ with respect to the barycenter $G$.

What is the largest volume of a sphere which touches to a unit sphere internally and touches externally to a regular tetrahedron whose corners are over the unit sphere? $\textbf{(A)}\ \frac13 \qquad\textbf{(B)}\ \frac14 \qquad\textbf{(C)}\ \frac12\left ( 1 - \frac1{\sqrt3} \right ) \qquad\textbf{(D)}\ \frac12\left ( \frac{2\sqrt2}{\sqrt3} - 1 \right ) \qquad\textbf{(E)}\ \text{None}$

The edge lengths of a tetrahedron are a, b, c, d, e, f, the areas of its faces are S1, S2, S3, S4, and its volume is V . Prove that 2 [S1 S2 S3 S4](1/2) > 3V [abcdef](1/6) this problem comes from: http://www.imomath.com/othercomp/jkasfvgkusa/MonMO99.pdf I was just wondering if someone could write it in LATEX form. [color=red]_____________________________________ EDIT by moderator: If you type[/color] [code]The edge lengths of a tetrahedron are $a, b, c, d, e, f,$ the areas of its faces are $S_1, S_2, S_3, S_4,$ and its volume is $V.$ Prove that $2 \sqrt{S_1 S_2 S_3 S_4} > 3V \sqrt[6]{abcdef}$[/code] [color=red]it shows up as:[/color] The edge lengths of a tetrahedron are $ a, b, c, d, e, f,$ the areas of its faces are $ S_1, S_2, S_3, S_4,$ and its volume is $ V.$ Prove that $ 2 \sqrt{S_1 S_2 S_3 S_4} > 3V \sqrt[6]{abcdef}$

Prove that the product of the sines of two opposite dihedrals in a tetrahedron is proportional to the product of the lengths of the edges of these dihedrals.

For a positive real number $a$, let $C$ be the cube with vertices at $(\pm a, \pm a, \pm a)$ and let $T$ be the tetrahedron with vertices at $(2a,2a,2a),(2a, -2a, -2a),(-2a, 2a, -2a),(-2a, -2a, -2a)$. If the intersection of $T$ and $C$ has volume $ka^3$ for some $k$, find $k$.

Consider the paper triangle whose vertices are $(0,0), (34,0),$ and $(16,24).$ The vertices of its midpoint triangle are the midpoints of its sides. A triangular pyramid is formed by folding the triangle along the sides of its midpoint triangle. What is the volume of this pyramid?

Four of the eight vertices of a cube are the vertices of a regular tetrahedron. Find the ratio of the surface area of the cube to the surface area of the tetrahedron. $\text{(A)} \ \sqrt 2 \qquad \text{(B)} \ \sqrt 3 \qquad \text{(C)} \ \sqrt{\frac{3}{2}} \qquad \text{(D)} \ \frac{2}{\sqrt{3}} \qquad \text{(E)} \ 2$

Let $ABCD$ be a tetrahedron and $A'$, $B'$, $C'$ be arbitrary points on the edges $[DA]$, $[DB]$, $[DC]$, respectively. One considers the points $P_c \in [AB]$, $P_a \in [BC]$, $P_b \in [AC]$ and $P'_c \in [A'B']$, $P'_a \in [B'C']$, $P'_b \in [A'C']$ such that $$\frac{P_cA}{P_cB}= \frac{P'_cA'}{P'_cB'}=\frac{AA'}{BB'}\,\,\, , \,\,\,\frac{P_aB}{P_aC}= \frac{P'_aB'}{P'_aC'}=\frac{BB'}{CC'}\,\,\, , \,\,\, \frac{P_bC}{P_bA}= \frac{P'_bC'}{P'_bA'}=\frac{CC'}{AA'}$$ Prove that: a) the lines $AP_a,$ $BP_b$, $CP_c$ have a common point $P$ and the lines $A'P'_a$, $B'P'_b$ , $C'P'_c$ have a common point $P'$ b) $\frac{PC}{PP_c}=\frac{P'C'}{P'P'_c} $ c) if $A', B', C'$ are variable points on the edges $[DA]$, $[DB]$, $[DC]$, then the line $PP'$ is always parallel to a fixed line.

Two triangles $ABC,ABD$ (with the common side $c=AB$) are given in space. Triangle $ABC$ is right with hypotenuse $AB$, $ABD$ is equilateral. Denote $\varphi$ the dihedral angle between planes $ABC,ABD$. 1) Determine the length of $CD$ in terms of $a=BC,b=CA,c$ and $\varphi$. 2) Determine all values of $\varphi$ such that the tetrahedron $ABCD$ has four sides of the same length.

Consider positive numbers $h, s_1, s_2$, and a spatial triangle $\vartriangle ABC$. How many ways are there to select a point $D$ so that the height of tetrahedron $ABCD$ drawn from $D$ equals $h$, and the areas of faces $ACD$ and $BCD$ equal $s_1$ and $s_2$, respectively?

Cube $ABCDEFGH$, labeled as shown below, has edge length $1$ and is cut by a plane passing through vertex $D$ and the midpoints $M$ and $N$ of $\overline{AB}$ and $\overline{CG}$ respectively. The plane divides the cube into two solids. The volume of the larger of the two solids can be written in the form $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$. [asy] draw((0,0)--(10,0)--(10,10)--(0,10)--cycle); draw((0,10)--(4,13)--(14,13)--(10,10)); draw((10,0)--(14,3)--(14,13)); draw((0,0)--(4,3)--(4,13), dashed); draw((4,3)--(14,3), dashed); dot((0,0)); dot((0,10)); dot((10,10)); dot((10,0)); dot((4,3)); dot((14,3)); dot((14,13)); dot((4,13)); dot((14,8)); dot((5,0)); label("A", (0,0), SW); label("B", (10,0), S); label("C", (14,3), E); label("D", (4,3), NW); label("E", (0,10), W); label("F", (10,10), SE); label("G", (14,13), E); label("H", (4,13), NW); label("M", (5,0), S); label("N", (14,8), E); [/asy]

Given a tetrahedron such that product of the opposite edges is $ 1$. Let the angle between the opposite edges be $ \alpha$, $ \beta$, $ \gamma$, and circumradii of four faces be $ R_1$, $ R_2$, $ R_3$, $ R_4$. Prove that \[ \sin^2\alpha \plus{} \sin^2\beta \plus{} \sin^2\gamma\ge\frac {1}{\sqrt {R_1R_2R_3R_4}} \]

Let $a,b$ be the lengths of two nonadjacent edges of a tetrahedron with inradius $r$. Prove that \[r<\frac{ab}{2(a+b)}.\]

Prove that the sum of the six angles subtended at an interior point of a tetrahedron by its six edges is greater than 540°.

In regular tetrahedron $ABCD$, $E\in AB,F\in CD$, satisfying: $\frac{|AE|}{|EB|}=\frac{|CF|}{|FD|}=\lambda(\lambda\in R_+)$. Note that $f(\lambda)=\alpha_{\lambda}+\beta_{\lambda}$, where $\alpha_{\lambda}=<EF,AC>,\alpha_{\lambda}=<EF,BD>$. $\text{(A)}$ $f(\lambda)$ increases in $(0,+\infty)$ $\text{(B)}$ $f(\lambda)$ decreases in $(0,+\infty)$ $\text{(C)}$ $f(\lambda)$ increases in $(0,1)$, decreases in $(1,+\infty)$ $\text{(D)}$ $f(\lambda)$ is a fixed value in $(0,+\infty)$

Let $ABCD$ be a tetrahedron and $O$ its incenter, and let the line $OD$ be perpendicular to $AD$. Find the angle between the planes $DOB$ and $DOC.$

Four faces of tetrahedron $ABCD$ are congruent triangles whose angles form an arithmetic progression. If the lengths of the sides of the triangles are $a < b < c$, determine the radius of the sphere circumscribed about the tetrahedron as a function on $a, b$, and $c$. What is the ratio $c/a$ if $R = a \ ?$

In the $[ABCD]$ tetrahedron having all the faces acute angled triangles, is denoted by $r_X$, $R_X$ the radius lengths of the circle inscribed and circumscribed respectively on the face opposite to the $X \in \{A,B,C,D\}$ peak, and with $R$ the length of the radius of the sphere circumscribed to the tetrahedron. Show that inequality occurs$$8R^2 \geq (r_A + R_A)^2 + (r_B + R_B)^2 + (r_C + R_C)^2 + (r_D + R_D)^2$$

Let a [i]lattice tetrahedron[/i] denote a tetrahedron whose vertices have integer coordinates. Given a lattice tetrahedron, a [i]move[/i] consists of picking some vertex and moving it parallel to one of the three edges of the face opposite the vertex so that it lands on a different point with integer coordinates. Prove that any two lattice tetrahedra with the same volume can be transformed into each other by a series of moves

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

[b](a)[/b] Given a tetrahedron $ABCD$ and its four altitudes (i.e., lines through each vertex, perpendicular to the opposite face), assume that the altitude dropped from $D$ passes through the orthocenter $H_4$ of $\triangle ABC$. Prove that this altitude $DH_4$ intersects all the other three altitudes. [b](b)[/b] If we further know that a second altitude, say the one from vertex A to the face $BCD$, also passes through the orthocenter $H_1$ of $\triangle BCD$, then prove that all four altitudes are concurrent and each one passes through the orthocenter of the respective triangle.

The medians from $ A $ to the faces $ ABC,ABD,ACD $ of a tetahedron $ ABCD $ are pairwise perpendicular. Show that the edges from $ A $ have equal lengths. [i]Dinu Șerbănescu[/i]