A set $ E$ of points in the 3D space let $ L(E)$ denote the set of all those points which lie on lines composed of two distinct points of $ E.$ Let $ T$ denote the set of all vertices of a regular tetrahedron. Which points are in the set $ L(L(T))?$

A paper equilateral triangle of side length $2$ on a table has vertices labeled $A,B,C.$ Let $M$ be the point on the sheet of paper halfway between $A$ and $C.$ Over time, point $M$ is lifted upwards, folding the triangle along segment $BM,$ while $A,B,$ and $C$ on the table. This continues until $A$ and $C$ touch. Find the maximum volume of tetrahedron $ABCM$ at any time during this process.

Given a tetrahedron $ABCD$, $E$, $F$, $G$, are on the respectively on the segments $AB$, $AC$ and $AD$. Prove that: i) area $EFG \leq$ max{area $ABC$,area $ABD$,area $ACD$,area $BCD$}. ii) The same as above replacing "area" for "perimeter".

Given a tetrahedron $ ABCD $ whose edges $ AB, BC, CD, DA $ are tangent to a certain sphere. Prove that the points of tangency lie in the same plane.

Determine the lengths of the edges of a right tetrahedron of volume $a^3$ so that the sum of its edges' lengths is minumum.

A sphere inscribed in a tetrahedron touches one face at the intersection of its angle bisectors, a second face at the intersection of its altitudes, and a third face at the intersection of its medians. Show that the tetrahedron is regular. [i]N. Agakhanov[/i]

Two regular tetrahedrons $A$ and $B$ are made with the 8 vertices of a unit cube. (this way is unique) What's the volume of $A\cup B$?

Let $ABCD$ be a tetrahedron in which the opposite sides are equal and form equal angles. Prove that it is regular.

Let be given an arbitrary tetrahedron $ABCD$ with volume $V$. Consider all lines which pass through the barycenter $S$ of the tetrahedron and intersect the edges $AD,BD,CD$ at points $A',B',C$ respectively. It is known that among the obtained tetrahedra there exists one with the minimal volume. Express this minimal volume in terms of $V$

A regular tetrahedron with an edge of $30$ cm rests on one of its faces. Assuming it is hollow, $2$ liters of water are poured into it. Find the height of the ''upper'' liquid and the area of the ''free'' surface of the water.

In a regular tetrahedron the centers of the four faces are the vertices of a smaller tetrahedron. The ratio of the volume of the smaller tetrahedron to that of the larger is $m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m+n.$

Let $ ABCD$ be a regular tetrahedron and $ M,N$ distinct points in the planes $ ABC$ and $ ADC$ respectively. Show that the segments $ MN,BN,MD$ are the sides of a triangle.

The edges of a regular tetrahedron with vertices $A ,~ B,~ C$, and $D$ each have length one. Find the least possible distance between a pair of points $P$ and $Q$, where $P$ is on edge $AB$ and $Q$ is on edge $CD$. $\textbf{(A) }\frac{1}{2}\qquad\textbf{(B) }\frac{3}{4}\qquad\textbf{(C) }\frac{\sqrt{2}}{2}\qquad\textbf{(D) }\frac{\sqrt{3}}{2}\qquad\textbf{(E) }\frac{\sqrt{3}}{3}$ [asy] size(150); import patterns; pair D=(0,0),C=(1,-1),B=(2.5,-0.2),A=(1,2),AA,BB,CC,DD,P,Q,aux; add("hatch",hatch()); //AA=new A and etc. draw(rotate(100,D)*(A--B--C--D--cycle)); AA=rotate(100,D)*A; BB=rotate(100,D)*D; CC=rotate(100,D)*C; DD=rotate(100,D)*B; aux=midpoint(AA--BB); draw(BB--DD); P=midpoint(AA--aux); aux=midpoint(CC--DD); Q=midpoint(CC--aux); draw(AA--CC,dashed); dot(P); dot(Q); fill(DD--BB--CC--cycle,pattern("hatch")); label("$A$",AA,W); label("$B$",BB,S); label("$C$",CC,E); label("$D$",DD,N); label("$P$",P,S); label("$Q$",Q,E); //Credit to TheMaskedMagician for the diagram [/asy]

Consider a tetrahedron $ABCD$. A point $X$ is chosen outside the tetrahedron so that segment $XD$ intersects face $ABC$ in its interior point. Let $A' , B'$ , and $C'$ be the projections of $D$ onto the planes $XBC, XCA$, and $XAB$ respectively. Prove that $A' B' + B' C' + C' A' \le DA + DB + DC$. (V.Yassinsky)

Areas of four surfaces of a tetrahedron are $S_1,S_2,S_3,S_4$. And the largest one of them is $S$. $\lambda=\frac{S_1+S_2+S_3+S_4}{S}$, then $\lambda$ always satisfies $\text{(A)}2<\lambda\leq4\qquad\text{(B)}3<\lambda<4\qquad\text{(C)}2.5<\lambda\leq4.5\qquad\text{(D)}3.5<\lambda<5.5$

The points of space are painted in two colors. Prove that there is a tetrahedron such that all its vertices and its centroid are of the same color.

Let $ABCD$ be a regular tetrahedron with side $a$. Take $E,E'$ on the edge $AB, F, F'$ on the edge $AC$ and $G,G'$ on the edge AD so that $AE =a/6,AE' = 5a/6,AF= a/4,AF'= 3a/4,AG = a/3,AG'= 2a/3$. Compute the volume of $EFGE'F'G'$ in term of $a$ and find the angles between the lines $AB,AC,AD$ and the plane $EFG$.

Prove that a tetrahedron with just one edge length greater than $1$ has volume at most $ \frac{1}{8}.$

Is there exist four points on plane, such that distance between any two of them is integer odd number? May be it is geometry or number theory or combinatoric, I don't know, so it here :blush:

[i]a)[/i] The solid $S$ is defined as the intersection of the six spheres with the six edges of a regular tetrahedron $T$, with edge length $1$, as diameters. Prove that $S$ contains two points at a distance $\frac{1}{\sqrt 6}.$ [i]b)[/i] Using the same assumptions in [i]a)[/i], prove that no pair of points in $S$ has a distance larger than $\frac{1}{\sqrt 6}.$

Let $ ABCD$ be a regular tetrahedron, and let $ O$ be the centroid of triangle $ BCD$. Consider the point $ P$ on $ AO$ such that $ P$ minimizes $ PA \plus{} 2(PB \plus{} PC \plus{} PD)$. Find $ \sin \angle PBO$.

Consider the solid with 4 triangles and 4 regular hexagons as faces, where each triangle borders 3 hexagons, and all the sides are of length 1. Compute the [i]square[/i] of the volume of the solid. Express your result in reduced fraction and concatenate the numerator with the denominator (e.g., if you think that the square is $\frac{1734}{274}$, then you would submit 1734274).

Six segments are such that any three can form a triangle. Is it true that these segments can be used to form a tetrahedron? (S. Markelov)

Let $ABCDEFGH$ be a cube. Consider a plane whose intersection with the tetrahedron $ABDE$ is a triangle with an obtuse angle $\varphi.$ Determine all $\varphi>\pi/2$ for which there is such a plane.

In space, a point $O$ and a finite set of vectors $ \overrightarrow{v_1},\ldots,\overrightarrow{v_n} $ are given . We consider the set of points $ P $ for which the vector $ \overrightarrow{OP} $can be represented as a sum $ a_1 \overrightarrow{v_1} + \ldots + a_n\overrightarrow{v_n} $with coefficients satisfying the inequalities $ 0 \leq a_i \leq 1 $ $( i = 1, 2, \ldots, n $). Decide whether this set can be a tetrahedron.