Found problems: 619
Given a point $M$ inside a right tetrahedron. Prove that at least one tetrahedron edge is seen from the $M$ in an angle, that has a cosine not greater than $-1/3$. (e.g. if $A$ and $B$ are the vertices, corresponding to that edge, $cos(\widehat{AMB}) \le -1/3$)
Given a tetrahedron $ABCD$, let $x = AB \cdot CD$, $y = AC \cdot BD$, and $z = AD \cdot BC$. Prove that there exists a triangle with edges $x, y, z.$
A tetrahedron is a polyhedron composed of four triangular faces. Faces $ABC$ and $BCD$ of a tetrahedron $ABCD$ meet at an angle of $\pi/6$. The area of triangle $\triangle ABC$ is $120$. The area of triangle $\triangle BCD$ is $80$, and $BC = 10$. What is the volume of the tetrahedron? We call the volume of a tetrahedron as one-third the area of it's base times it's height.
Prove that for every point $M$ on the surface of a regular tetrahedron there exists a point $M'$ such that there are at least three different curves on the surface joining $M$ to $M'$ with the smallest possible length among all curves on the surface joining $M$ to $M'$.
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.
A convex polyhedron has no diagonals (every pair of vertices are connected by an edge). Prove that it is a tetrahedron.
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 $G$.
$(CZS 1)$ Given a unit cube, find the locus of the centroids of all tetrahedra whose vertices lie on the sides of the cube.
Let $P$ be a point inside a regular tetrahedron ABCD with edge length $1$. Show that $$d(P,AB)+d(P,AC)+d(P,AD)+d(P,BC)+d(P,BD)+d(P,CD) \ge \frac{3}{2} \sqrt2$$ , with equality only when $P$ is the centroid of $ABCD$.
Here $d(P,XY)$ denotes the distance from point $P$ to line $XY$.
Let $K$ and $M$ be points on the side $AB$ of a triangle $\triangle{ABC}$, and let $L$ and $N$ be points on the side $AC$. The point $K$ is between $M$ and $B$, and the point $L$ is between $N$ and $C$. If $\frac{BK}{KM}=\frac{CL}{LN}$, then prove that the orthocentres of the triangles $\triangle{ABC}$, $\triangle{AKL}$ and $\triangle{AMN}$ lie on one line.
In the space is given a tetrahedron with length of the edge $2$. Prove that distances from some point $M$ to all of the vertices of the tetrahedron are integer numbers if and only if $M$ is a vertex of tetrahedron.
[i]J. Tabov[/i]
Suppose $\mathcal{T}=A_0A_1A_2A_3$ is a tetrahedron with $\angle A_1A_0A_3 = \angle A_2A_0A_1 = \angle A_3A_0A_2 = 90^\circ$, $A_0A_1=5, A_0A_2=12$ and $A_0A_3=9$. A cube $A_0B_0C_0D_0E_0F_0G_0H_0$ with side length $s$ is inscribed inside $\mathcal{T}$ with $B_0\in \overline{A_0A_1}, D_0 \in \overline{A_0A_2}, E_0 \in \overline{A_0A_3}$, and $G_0\in \triangle A_1A_2A_3$; what is $s$?
Let $P_1P_2P_3P_4$ be a tetrahedron in $\mathbb{R}^3$ and let $O$ be a point equidistant from each of its vertices. Suppose there exists a point $H$ such that for each $i$, the line $P_iH$ is perpendicular to the plane through the other three vertices. Line $P_1H$ intersects the plane through $P_2, P_3, P_4$ at $A$, and contains a point $B\neq P_1$ such that $OP_1=OB$. Show that $HB=3HA$.
[i]Michael Ren[/i]
$ABCD$ is a regular tetrahedron with side length $1$. Find the area of the cross section of $ABCD$ cut by the plane that passes through the midpoints of $AB$, $AC$, and $CD$.
A solid tetrahedron is sliced off a solid wooden unit cube by a plane passing through two nonadjacent vertices on one face and one vertex on the opposite face not adjacent to either of the first two vertices. The tetrahedron is discarded and the remaining portion of the cube is placed on a table with the cut surface face down. What is the height of this object?
$ \textbf{(A)}\ \dfrac{\sqrt{3}}{3}\qquad\textbf{(B)}\ \dfrac{2\sqrt{2}}{3}\qquad\textbf{(C)}\ 1\qquad\textbf{(D)}\ \dfrac{2\sqrt{3}}{3}\qquad\textbf{(E)}\ \sqrt{2} $
Prove that the perpendiculars, drawn from the midpoints of the edges of the base of a given tetrahedron to the opposite lateral edges, have a common point if and only if the circumcenter of the tetrahedron, the centroid of the base, and the top vertex of the tetrahedron are collinear.
A tetrahedron is inscribed in a sphere of radius $1$ such that the center of the sphere is inside the tetrahedron. Prove that the sum of lengths of all edges of the tetrahedron is greater than 6.
A real number $f(X)\neq 0$ is assigned to each point $X$ in the space.
It is known that for any tetrahedron $ABCD$ with $O$ the center of the inscribed sphere, we have :
\[ f(O)=f(A)f(B)f(C)f(D). \]
Prove that $f(X)=1$ for all points $X$.
[i]Proposed by Aleksandar Ivanov[/i]
Points $A, B, C, D$ lie on the sphere of radius $1$. It is known that $AB\cdot AC\cdot AD\cdot BC\cdot BD\cdot CD=\frac{512}{27}$. Prove that $ABCD$ is a regular tetrahedron.
A tetrahedron is to be cut by three planes which form a parallelepiped whose three faces and all vertices lie on the surface of the tetrahedron.
(a) Can this be done so that the volume of the parallelepiped is at least $ \frac{9}{40}$ of the volume of the tetrahedron?
(b) Determine the common point of the three planes if the volume of the parallelepiped is $ \frac{11}{50}$ of the volume of the tetrahedron.
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]
The areas of the faces $ABD,ACD,BCD,BCA$ of a tetrahedron $ABCD$ are $S_1,S_2,Q_1,Q_2$, respectively. The angle between the faces $ABD$ and $ACD$ equals $\alpha$, and the angle between $BCD$ and $BCA$ is $\beta$. Prove that
$$S_1^2+S_2^2-2S_1S_2\cos\alpha=Q_1^2+Q_2^2-2Q_1Q_2\cos\beta.$$
Each face of a tetrahedron can be placed in a circle of radius $ 1$. Show that the tetrahedron can be placed in a sphere of radius $ \frac{3}{2\sqrt2}$.
Let $OABC$ be a tetrahedon with $\angle BOC=\alpha,\angle COA =\beta$ and $\angle AOB =\gamma$. The angle between the faces $OAB$ and $OAC$ is $\sigma$ and the angle between the faces $OAB$ and $OBC$ is $\rho$.
Show that $\gamma > \beta \cos\sigma + \alpha \cos\rho$.
In tetrahedron $ABCD$ with circumradius $2$, $AB = 2$, $CD = \sqrt{7}$, and $\angle ABC = \angle BAD = \frac{\pi}{2}$. Find all possible angles between the planes containing $ABC$ and $ABD$.