Found problems: 619
Through the point of intersection of the medians of each of the faces a tetrahedron is drawn perpendicular to this face. Prove that all these four lines intersect at one point if and only if the four lines containing the heights of this tetrahedron intersect at one point .
Prove that if two segments of a tetrahedron, going from the ends of some edge to the centers of the inscribed circles of opposite faces, intersect, then the segments issued from the ends of the crossing with it edges to the centers of the inscribed circles of the other two faces, also intersect.
Show that the perimeter of an arbitrary planar section of a tetrahedron is less than the perimeter of one of the faces of the tetrahedron. [Gy. Hajos]
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)
A sphere $S$ lies within tetrahedron $ABCD$, touching faces $ABD, ACD$, and $BCD$, but having no point in common with plane $ABC$. Let $E$ be the point in the interior of the tetrahedron for which $S$ touches planes $ABE$, $ACE$, and $BCE$ as well. Suppose the line $DE$ meets face $ABC$ at $F$, and let $L$ be the point of $S$ nearest to plane $ABC$. Show that segment $FL$ passes through the centre of the inscribed sphere of tetrahedron $ABCE$.
KöMaL A.723. (April 2018), G. Kós
By a proper divisor of a natural number we mean a positive integral divisor other than 1 and the number itself. A natural number greater than 1 will be called "nice" if it is equal to the product of its distinct proper divisors. What is the sum of the first ten nice numbers?
Each face of a tetrahedron is a triangle with sides $a, b,$c and the tetrahedon has circumradius 1. Find $a^2 + b^2 + c^2$.
Let $ABCD$ be a tetrahedron and $M$ and $N$ be the midpoints of $AC$ and $BD$, respectively. Show that for every point $P \in (MN)$ with $P \neq M$ and $P \neq N$, there exist unique points $X$ and $Y$ on segments $AB$ and $CD$, respectively, such that $X,P,Y$ are collinear.
$ABCD$ is a tetrahedron such that we can find a sphere $k(A,B,C)$ through $A, B, C$ which meets the plane $BCD$ in the circle diameter $BC$, meets the plane $ACD$ in the circle diameter $AC$, and meets the plane $ABD$ in the circle diameter $AB$. Show that there exist spheres $k(A,B,D)$, $k(B,C,D)$ and $k(C,A,D)$ with analogous properties.
Given an acute triangle $T$ with sides $a,b,c$, find the tetrahedra with base $T$ whose all faces are acute triangles of the same area.
Prove that if all dihedral angles of a tetrahedron are acute, then all its faces are acute-angled triangles.
In a tetrahedron $ABCD$, all angles at vertex $D$ are equal to $\alpha$ and all dihedral angles between faces having $D$ as a vertex are equal to $\phi$. Prove that there exists a unique $\alpha$ for which $\phi=2\alpha$.
A tetrahedron is such that the center of the its circumscribed sphere is inside the tetrahedron.
Show that at least one of its edges has a size larger than or equal to the size of the edge of a regular tetrahedron inscribed in this same sphere.
A tetrahedron $T$ is inside a cube $C$. Prove that the volume of $T$ is at most one-third the volume of $C$.
Given a triangle, its midpoint triangle is obtained by joining the midpoints of its sides. A sequence of polyhedra $P_{i}$ is defined recursively as follows: $P_{0}$ is a regular tetrahedron whose volume is 1. To obtain $P_{i+1}$, replace the midpoint triangle of every face of $P_{i}$ by an outward-pointing regular tetrahedron that has the midpoint triangle as a face. The volume of $P_{3}$ is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
$P$ is a point inside the triangle $ABC$. The perpendicular distances from $P$ to the three sides have product $p$. Show that $p \le \frac{ 8 S^3}{27abc}$, where $S =$ area $ABC$ and $a, b, c$ are the sides. Prove a similar result for a tetrahedron.
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 $k$.
A tetrahedron is called a [i]MEMO-tetrahedron[/i] if all six sidelengths are different positive integers where one of them is $ 2$ and one of them is $ 3$. Let $ l(T)$ be the sum of the sidelengths of the tetrahedron $ T$.
(a) Find all positive integers $ n$ so that there exists a MEMO-Tetrahedron $ T$ with $ l(T)\equal{}n$.
(b) How many pairwise non-congruent MEMO-tetrahedrons $ T$ satisfying $ l(T)\equal{}2007$ exist? Two tetrahedrons are said to be non-congruent if one cannot be obtained from the other by a composition of reflections in planes, translations and rotations. (It is not neccessary to prove that the tetrahedrons are not degenerate, i.e. that they have a positive volume).
Consider a polyhedron with at least five faces such that exactly three edges emerge from each of its vertices. Two players play the following game: Each, in turn, signs his or her name on a previously unsigned face. The winner is the player who first succeeds in signing three faces that share a common vertex. Show that the player who signs first will always win by playing as well as possible.
Let $S$ be the set of all tetrahedra which satisfy:
(1) the base has area $1$,
(2) the total face area is $4$, and
(3) the angles between the base and the other three faces are all equal.
Find the element of $S$ which has the largest volume.
Let $ABCD$ be a tetrahedron and let $H_{a},H_{b},H_{c},H_{d}$ be the orthocenters of triangles $BCD,CDA,DAB,ABC$, respectively.
Prove that lines $AH_{a},BH_{b},CH_{c}, DH_{d}$ are concurrent
if and only if
$AB^2 + CD^2 = AC^2 + BD^2 = AD^2 + BC^2$
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))?$
The insphere and the exsphere opposite to the vertex $D$ of a (not necessarily regular) tetrahedron $ABCD$ touch the face $ABC$ in the points $X$ and $Y$, respectively. Show that $\measuredangle XAB=\measuredangle CAY$.
Prove that the orthogonal projections of the vertex $ D $ of the tetrahedron $ ABCD $ onto the bisectors of the internal and external dihedral angles at the edges $ \overline{AB} $, $ \overline{BC} $ and $ \overline{CA} $ belong to one plane .
Let $ABCD$ be a regular tetrahedron with side length $2$. The plane parallel to edges $AB$ and $CD$ and lying halfway between them cuts $ABCD$ into two pieces. Find the surface area of one of these pieces.