Found problems: 619
For $n \in \mathbb{N}$, consider in $\mathbb{R}^3$ the regular tetrahedron with vertices $O(0, 0, 0)$, $A(n, 9n, 4n)$, $B(9n, 4n, n)$ and $C(4n, n, 9n)$. Show that the number $N$ of points $(x, y, z)$, $[x, y, z \in \mathbb{Z}]$ inside or on the boundary of the tetrahedron $OABC$ is given by$$N=\frac{343n^3}{3}+\frac{35n^2}{2}+\frac{7n}{6}+1$$
Prove that a tetrahedron with just one edge length greater than $1$ has volume at most $ \frac{1}{8}.$
Show that it is impossible to dissect an arbitary tetrahedron into six parts by planes or portions thereof so that each of the parts has a plane of symmetry.
Prove that if the four faces of a tetrahedron are of the same area they are equal.
In the tetrahedron $ABCD$, $E$ and $F$ are the midpoints of $BC$ and $AD$, $G$ is the midpoint of the segment $EF$. Construct a plane through $G$ intersecting the segments $AB$, $AC$, $AD$ in the points $M,N,P$ respectively in such a way that the sum of the volumes of the tetrahedrons $BMNP$, $CMNP$ and $DMNP$ to be minimal.
[i]H. Lesov[/i]
Prove the theorem:
In a tetrahedron, the plane bisector of any dihedral angle divides the opposite edge into segments proportional to the areas of the tetrahedron faces that form this dihedral angle.
Let $ABCD$ be a tetrahedron with $BA \perp AC,DB \perp (BAC)$. Denote by $O$ the midpoint of $AB$, and $K$ the foot of the perpendicular from $O$ to $DC$. Suppose that $AC = BD$. Prove that $\frac{V_{KOAC}}{V_{KOBD}}=\frac{AC}{BD}$ if and only if $2AC \cdot BD = AB^2$.
Prove that the sums of the opposite dihedral angles of a tetrahedron are equal if and only if the sums of the opposite edges of this tetrahedron are equal.
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 $(0,+\infty)$
In the tetrahedron $ABCD$ the radius of its inscribed sphere is $r$ and the radiuses of the exinscribed spheres (each tangent with a face of the tetrahedron and with the planes of the other faces) are $r_A, r_B, r_C, r_D.$ Prove the inequality $$\frac{1}{\sqrt{r_A^2-r_Ar_B+r_B^2}}+\frac{1}{\sqrt{r_B^2-r_Br_C+r_C^2}}+\frac{1}{\sqrt{r_C^2-r_Cr_D+r_D^2}}+\frac{1}{\sqrt{r_D^2-r_Dr_A+r_A^2}}\leq\frac{2}{r}.$$
Given a triangle $ABC$. Consider all the tetrahedrons $PABC$ with $PH$ -- the smallest of all tetrahedron's heights. Describe the set of all possible points $H$.
Let $ABCD$ be a random tetrahedron. Let $E$ and $F$ be the midpoints of segments $AB$ and $CD$, respectively. If the angle $a$ is between $AD$ and $BC$, determine $cos a$ in terms of $EF, AD$ and $BC$.
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".
Let $ABCD$ be a tetrahedron having each sum of opposite sides equal to $1$. Prove that
\[r_A + r_B + r_C + r_D \leq \frac{\sqrt 3}{3}\]
where $r_A, r_B, r_C, r_D$ are the inradii of the faces, equality holding only if $ABCD$ is regular.
A variable tetrahedron $ABCD$ has the following properties:
Its edge lengths can change as well as its vertices, but the opposite edges remain equal $(BC = DA, CA = DB, AB = DC)$; and the vertices $A,B,C$ lie respectively on three fixed spheres with the same center $P$ and radii $3, 4, 12$. What is the maximal length of $PD$?
Prove that the intersection of a plane and a regular tetrahedron can be an obtuse-angled triangle and that the obtuse angle in any such triangle is always smaller than $ 120^{\circ}.$
In a tetrahedron we know that sum of angles of all vertices is $180^\circ.$ (e.g. for vertex $A$, we have $\angle BAC + \angle CAD + \angle DAB=180^\circ.$)
Prove that faces of this tetrahedron are four congruent triangles.
In the tetrahedron $SABC $ at the height $SH$ the following point $O$ is chosen, such that: $$\angle AOS + \alpha = \angle BOS + \beta = \angle COS + \gamma = 180^o, $$ where $\alpha, \beta, \gamma$ are dihedral angles at the edges $BC, AC, AB $, respectively, at this point $H$ lies inside the base $ABC$. Let ${{A} _ {1}}, \, {{B} _ {1}}, \, {{C} _ {1}} $be the points of intersection of lines and planes: ${{A} _ {1}} = AO \cap SBC $, ${{B} _ {1}} = BO \cap SAC $, ${{C} _ {1}} = CO \cap SBA$ . Prove that if the planes $ABC $ and ${{A} _ {1}} {{B} _ {1}} {{C} _ {1}} $ are parallel, then $SA = SB = SC $.
(Alexey Klurman)
Show how to divide space into
(a) congruent tetrahedra,
(b) congruent “equifaced” tetrahedra.
(A tetrahedron is called equifaced if all its faces are congruent triangles.)
(NB Vassiliev)
Given a trihedral angle Sxyz. A plane (P) not through S cuts Sx,Sy,Sz respectively at A,B,C. On the plane (P), outside triangle ABC, construct triangles DAB,EBC,FCA which are confruent to the triangles SAB,SBC,SCA respectively. Let (T) be the sphere lying inside Sxyz, but not inside the tetrahedron SABC, toucheing the planes containing the faces of SABC. Prove that (T) touches the plane (P) at the circumcenter of triangle DEF.
Let $OABC$ be a tetrahedron such that $\angle AOB = \angle BOC = \angle COA = 90^\circ$ and its faces have integral surface areas. If $[OAB] = 20$ and $[OBC] = 14$, find the sum of all possible values of $[OCA][ABC]$. (Here $[\triangle]$ denotes the area of $\triangle$.)
[i]Proposed by Robin Park[/i]
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 $DOC.$
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]
In a given tedrahedron $ ABCD$ let $ K$ and $ L$ be the centres of edges $ AB$ and $ CD$ respectively. Prove that every plane that contains the line $ KL$ divides the tedrahedron into two parts of equal volume.
In a tetrahedron $ABCD, E$ and $F$ are the midpoints of the medians from $A$ and $D$. Find the ratio of the volumes of tetrahedra $BCEF$ and $ABCD$.
Note: Median in a tetrahedron connects a vertex and the centroid of the opposite side.