Found problems: 619
2008 Flanders Math Olympiad, 3
A quadrilateral pyramid and a regular tetrahedron have edges that are all equal in length. They are glued together so that they have in common $1$ equilateral triangle . Prove that the resulting body has exactly $5$ sides.
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?
1988 IMO Longlists, 8
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.
2005 Swedish Mathematical Competition, 6
A regular tetrahedron of edge length $1$ is orthogonally projected onto a plane. Find the largest possible area of its image.
1976 Bulgaria National Olympiad, Problem 3
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]
1984 IMO Longlists, 11
Prove that the volume of a tetrahedron inscribed in a right circular cylinder of volume $1$ does not exceed $\frac{2}{3 \pi}.$
2001 Tournament Of Towns, 5
Nine points are drawn on the surface of a regular tetrahedron with an edge of $1$ cm. Prove that among these points there are two located at a distance (in space) no greater than $0.5$ cm.
2019 Jozsef Wildt International Math Competition, W. 59
In the any $[ABCD]$ tetrahedron we denote with $\alpha$, $\beta$, $\gamma$ the measures, in radians, of the angles of the three pairs of opposite edges and with $r$, $R$ the lengths of the rays of the sphere inscribed and respectively circumscribed the tetrahedron. Demonstrate inequality$$\left(\frac{3r}{R}\right)^3\leq \sin \frac{\alpha +\beta +\gamma}{3}$$(A refinement of inequality $R \geq 3r$).
1990 IMO Longlists, 61
Prove that we can fill in the three dimensional space with regular tetrahedrons and regular octahedrons, all of which have the same edge-lengths. Also find the ratio of the number of the regular tetrahedrons used and the number of the regular octahedrons used.
2003 All-Russian Olympiad Regional Round, 11.7
Given a tetrahedron $ABCD.$ The sphere $\omega$ inscribed in it touches the face $ABC$ at point $T$. Sphere $\omega' $ touches face $ABC$ at point $T'$ and extensions of faces $ABD$, $BCD$, $CAD$. Prove that the lines $AT$ and $AT'$ are symmetric wrt bisector of angle $\angle BAC$
1968 Yugoslav Team Selection Test, Problem 6
Prove that the incenter coincides with the circumcenter of a tetrahedron if and only if each pair of opposite edges are of equal length.
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) ?
1988 Iran MO (2nd round), 2
In tetrahedron $ABCD$ let $h_a, h_b, h_c$ and $h_d$ be the lengths of the altitudes from each vertex to the opposite side of that vertex. Prove that
\[\frac{1}{h_a} <\frac{1}{h_b}+\frac{1}{h_c}+\frac{1}{h_d}.\]
1999 Romania National Olympiad, 2
On the sides $(AB)$, $(BC)$, $(CD)$ and $(DA)$ of the regular tetrahedron $ABCD$, one considers the points $M$, $N$, $P$, $Q$, respectively Prove that
$$MN \cdot NP \cdot PQ \cdot QM \ge AM \cdot BN \cdot CP \cdot DQ.$$
2000 Harvard-MIT Mathematics Tournament, 32
How many (nondegenerate) tetrahedrons can be formed from the vertices of an $n$-dimensional hypercube?
2011 Bundeswettbewerb Mathematik, 4
Let $ABCD$ be a tetrahedron that is not degenerate and not necessarily regular, where sides $AD$ and $BC$ have the same length $a$, sides $BD$ and $AC$ have the same length $b$, side $AB$ has length $c_1$ and the side $CD$ has length $c_2$. There is a point $P$ for which the sum of the distances to the vertices of the tetrahedron is minimal. Determine this sum depending on the quantities $a, b, c_1$ and $c_2$.
1969 IMO Longlists, 58
$(SWE 1)$ Six points $P_1, . . . , P_6$ are given in $3-$dimensional space such that no four of them lie in the same plane. Each of the line segments $P_jP_k$ is colored black or white. Prove that there exists one triangle $P_jP_kP_l$ whose edges are of the same color.
1954 Polish MO Finals, 5
Prove that if in a tetrahedron $ ABCD $ opposite edges are equal, i.e. $ AB = CD $, $ AC = BD $, $ AD = BC $, then the lines passing through the midpoints of opposite edges are mutually perpendicular and are the axes of symmetry of the tetrahedron.
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 $m+n+k.$
1983 Polish MO Finals, 6
Prove that if all dihedral angles of a tetrahedron are acute, then all its faces are acute-angled triangles.
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 $FDEABC$.
1984 National High School Mathematics League, 8
Lengths of five edges of a tetrahedron are $1$, while the last one is $x$. Its volume is $F(x)$. On its domain of definition, we have
$\text{(A)}$ $F(x)$ is an increasing function, it has no maximum value.
$\text{(B)}$ $F(x)$ is an increasing function, it has maximum value.
$\text{(C)}$ $F(x)$ is not an increasing function, it has no maximum value.
$\text{(D)}$ $F(x)$ is an increasing function, it has maximum value.
2005 Romania Team Selection Test, 3
Prove that if the distance from a point inside a convex polyhedra with $n$ faces to the vertices of the polyhedra is at most 1, then the sum of the distances from this point to the faces of the polyhedra is smaller than $n-2$.
[i]Calin Popescu[/i]
1968 Polish MO Finals, 3
In a tetrahedron $ ABCD $ the edges $AD$, $ BD $, $ CD $ are equal. $ ABC $ Non-collinear points are chosen in the plane. $ A_1$, $B_1$, $C_1 $ The lines $DA_1$, $DB_1$, $DC_1 $ intersect the surface of the sphere circumscribed about the tetrahedron at points $ A_2$, $B_2$, $C_2 $, different from the point $ D $. Prove that the points $A_1$, $B_1$, $C_1$, $A_2$, $B_2$, $C_2$ lie on the surface of a certain sphere.
1956 Putnam, B3
A sphere is inscribed in a tetrahedron and each point of contact of the sphere with the four faces is joined to the vertices of the face containing the point. Show that the four sets of three angles so formed are identical.