Found problems: 619
1952 Poland - Second Round, 6
Prove that a plane that passes:
a) through the centers of two opposite edges of the tetrahedron and
b) through the center of one of the other edges of the tetrahedron
divides the tetrahedron into two parts of equal volumes.
Will the thesis remain true if we reject assumption (b) ?
1991 Arnold's Trivium, 86
Through the centre of a cube (tetrahedron, icosahedron) draw a straight line in such a way that the sum of the squares of its distances from the vertices is a) minimal, b) maximal.
1998 All-Russian Olympiad Regional Round, 11.7
Given two regular tetrahedrons with edges of length $\sqrt2$, transforming into one another with central symmetry. Let $\Phi$ be the set the midpoints of segments whose ends belong to different tetrahedrons. Find the volume of the figure $\Phi$.
1983 USAMO, 4
Six segments $S_1, S_2, S_3, S_4, S_5,$ and $S_6$ are given in a plane. These are congruent to the edges $AB, AC, AD, BC, BD,$ and $CD$, respectively, of a tetrahedron $ABCD$. Show how to construct a segment congruent to the altitude of the tetrahedron from vertex $A$ with straight-edge and compasses.
2017 Adygea Teachers' Geometry Olympiad, 4
A regular tetrahedron $SABC$ of volume $V$ is given. The midpoints $D$ and $E$ are taken on $SA$ and $SB$ respectively and the point $F$ is taken on the edge $SC$ such that $SF: FC = 1: 3$. Find the volume of the pentahedron $FDEABC$.
1981 Romania Team Selection Tests, 2.
Consider a tetrahedron $OABC$ with $ABC$ equilateral. Let $S$ be the area of the triangle of sides $OA$, $OB$ and $OC$. Show that $V\leqslant \dfrac12 RS$ where $R$ is the circumradius and $V$ is the volume of the tetrahedron.
[i]Stere Ianuș[/i]
2003 Tournament Of Towns, 6
Let $O$ be the center of insphere of a tetrahedron $ABCD$. The sum of areas of faces $ABC$ and $ABD$ equals the sum of areas of faces $CDA$ and $CDB$. Prove that $O$ and midpoints of $BC, AD, AC$ and $BD$ belong to the same plane.
2000 AIME Problems, 12
The points $A, B$ and $C$ lie on the surface of a sphere with center $O$ and radius 20. It is given that $AB=13, BC=14, CA=15,$ and that the distance from $O$ to triangle $ABC$ is $\frac{m\sqrt{n}}k,$ where $m, n,$ and $k$ are positive integers, $m$ and $k$ are relatively prime, and $n$ is not divisible by the square of any prime. Find $m+n+k.$
1985 AIME Problems, 12
Let $A$, $B$, $C$, and $D$ be the vertices of a regular tetrahedron, each of whose edges measures 1 meter. A bug, starting from vertex $A$, observes the following rule: at each vertex it chooses one of the three edges meeting at that vertex, each edge being equally likely to be chosen, and crawls along that edge to the vertex at its opposite end. Let $p = n/729$ be the probability that the bug is at vertex $A$ when it has crawled exactly 7 meters. Find the value of $n$.
2009 Tournament Of Towns, 3
Every edge of a tetrahedron is tangent to a given sphere. Prove that the three line segments joining the points of tangency of the three pairs of opposite edges of the tetrahedron are concurrent.
[i](7 points)[/i]
2011 Sharygin Geometry Olympiad, 3
Given two tetrahedrons $A_1A_2A_3A_4$ and $B_1B_2B_3B_4$. Consider six pairs of edges $A_iA_j$ and $B_kB_l$, where ($i, j, k, l$) is a transposition of numbers ($1, 2, 3, 4$) (for example $A_1A_2$ and $B_3B_4$). It is known that for all but one such pairs the edges are perpendicular. Prove that the edges in the remaining pair also are perpendicular.
2004 Harvard-MIT Mathematics Tournament, 7
We have a polyhedron such that an ant can walk from one vertex to another, traveling only along edges, and traversing every edge exactly once. What is the smallest possible total number of vertices, edges, and faces of this polyhedron?
2006 MOP Homework, 4
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
$AB^2 + CD^2 = AC^2 + BD^2 = AD^2 + BC^2$
1973 IMO Longlists, 7
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 the side lengths $x, y$ and $z$.
2000 Harvard-MIT Mathematics Tournament, 1
How many different ways are there to paint the sides of a tetrahedron with exactly $4$ colors? Each side gets its own color, and two colorings are the same if one can be rotated to get the other.
1998 Poland - Second Round, 6
Prove that the edges $AB$ and $CD$ of a tetrahedron $ABCD$ are perpendicular if and only if there exists a parallelogram $CDPQ$ such that $PA = PB = PD$ and $QA = QB = QC$.
2006 Polish MO Finals, 2
Tetrahedron $ABCD$ in which $AB=CD$ is given. Sphere inscribed in it is tangent to faces $ABC$ and $ABD$ respectively in $K$ and $L$. Prove that if points $K$ and $L$ are centroids of faces $ABC$ and $ABD$ then tetrahedron $ABCD$ is regular.
2005 Sharygin Geometry Olympiad, 11.6
The sphere inscribed in the tetrahedron $ABCD$ touches its faces at points $A',B',C',D'$. The segments $AA'$ and $BB'$ intersect, and the point of their intersection lies on the inscribed sphere. Prove that the segments $CC'$ and $DD'$ also intersect on the inscribed sphere.
1969 IMO Shortlist, 12
$(CZS 1)$ Given a unit cube, find the locus of the centroids of all tetrahedra whose vertices lie on the sides of the cube.
1968 Poland - Second Round, 5
The tetrahedrons $ ABCD $ and $ A_1B_1C_1D_1 $ are situated so that the midpoints of the segments $ AA_1 $, $ BB_1 $, $ CC_1 $, $ DD_1 $ are the centroids of the triangles $BCD$, $ ACD $, $ A B D $ and $ ABC $, respectively. What is the ratio of the volumes of these tetrahedrons?
1970 IMO Shortlist, 3
In the tetrahedron $ABCD,\angle BDC=90^o$ and the foot of the perpendicular from $D$ to $ABC$ is the intersection of the altitudes of $ABC$. Prove that: \[ (AB+BC+CA)^2\le6(AD^2+BD^2+CD^2). \] When do we have equality?
2008 Baltic Way, 5
Suppose that Romeo and Juliet each have a regular tetrahedron to the vertices of which some positive real numbers are assigned. They associate each edge of their tetrahedra with the product of the two numbers assigned to its end points. Then they write on each face of their tetrahedra the sum of the three numbers associated to its three edges. The four numbers written on the faces of Romeo's tetrahedron turn out to coincide with the four numbers written on Juliet's tetrahedron. Does it follow that the four numbers assigned to the vertices of Romeo's tetrahedron are identical to the four numbers assigned to the vertices of Juliet's tetrahedron?
2010 Paenza, 6
In space are given two tetrahedra with the same barycenter such that one of them contains the other.
For each tetrahedron, we consider the octahedron whose vertices are the midpoints of the tetrahedron's edges.
Prove that one of this octahedra contains the other.
1988 China National Olympiad, 5
Given three tetrahedrons $A_iB_i C_i D_i$ ($i=1,2,3$), planes $\alpha _i,\beta _i,\gamma _i$ ($i=1,2,3$) are drawn through $B_i ,C_i ,D_i$ respectively, and they are perpendicular to edges $A_i B_i, A_i C_i, A_i D_i$ ($i=1,2,3$) respectively. Suppose that all nine planes $\alpha _i,\beta _i,\gamma _i$ ($i=1,2,3$) meet at a point $E$, and points $A_1,A_2,A_3$ lie on line $l$. Determine the intersection (shape and position) of the circumscribed spheres of the three tetrahedrons.
2001 Croatia National Olympiad, Problem 2
A piece of paper in the shape of a square $FBHD$ with side $a$ is given. Points $G,A$ on $FB$ and $E,C$ on $BH$ are marked so that $FG=GA=AB$ and $BE=EC=CH$. The paper is folded along $DG,DA,DC$ and $AC$ so that $G$ overlaps with $B$, and $F$ and $H$ overlap with $E$. Compute the volume of the obtained tetrahedron $ABCD$.