Found problems: 619
1978 Polish MO Finals, 6
Prove that if $h_1,h_2,h_3,h_4$ are the altitudes of a tetrahedron and $d_1,d_2,d_3$ the distances between the pairs of opposite edges of the tetrahedron, then
$$\frac{1}{h_1^2}
+\frac{1}{h_2^2}
+\frac{1}{h_3^2}
+\frac{1}{h_4^2}
=\frac{1}{d_1^2}
+\frac{1}{d_2^2}
+\frac{1}{d_3^2}.$$
1993 Baltic Way, 20
Let $ \mathcal Q$ be a unit cube. We say that a tetrahedron is [b]good[/b] if all its edges are equal and all of its vertices lie on the boundary of $ \mathcal Q$. Find all possible volumes of good tetrahedra.
1983 IMO Longlists, 67
The altitude from a vertex of a given tetrahedron intersects the opposite face in its orthocenter. Prove that all four altitudes of the tetrahedron are concurrent.
1991 IMTS, 4
Let $a,b,c,d$ be the areas of the triangular faces of a tetrahedron, and let $h_a, h_b, h_c, h_d$ be the corresponding altitudes of the tetrahedron. If $V$ denotes the volume of tetrahedron, prove that
\[ (a+b+c+d)(h_a+h_b+h_c+h_d) \geq 48V \]
2007 All-Russian Olympiad, 7
Given a tetrahedron $ T$. Valentin wants to find two its edges $ a,b$ with no common vertices so that $ T$ is covered by balls with diameters $ a,b$. Can he always find such a pair?
[i]A. Zaslavsky[/i]
1989 Nordic, 2
Three sides of a tetrahedron are right-angled triangles having the right angle at their common vertex. The areas of these sides are $A, B$, and $C$. Find the total surface area of the tetrahedron.
2000 Iran MO (2nd round), 2
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.
1986 IMO Longlists, 80
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 $DOC.$
2015 Sharygin Geometry Olympiad, P23
A tetrahedron $ABCD$ is given. The incircles of triangles $ ABC$ and $ABD$ with centers $O_1, O_2$, touch $AB$ at points $T_1, T_2$. The plane $\pi_{AB}$ passing through the midpoint of $T_1T_2$ is perpendicular to $O_1O_2$. The planes $\pi_{AC},\pi_{BC}, \pi_{AD}, \pi_{BD}, \pi_{CD}$ are defined similarly. Prove that these six planes have a common point.
1987 Poland - Second Round, 2
Prove that the sum of the plane angles at each of the vertices of a given tetrahedron is $ 180^{\circ} $ if and only if all its faces are congruent.
1998 All-Russian Olympiad, 7
A tetrahedron $ABCD$ has all edges of length less than $100$, and contains two nonintersecting spheres of diameter $1$. Prove that it contains a sphere of diameter $1.01$.
1978 USAMO, 4
(a) Prove that if the six dihedral (i.e. angles between pairs of faces) of a given tetrahedron are congruent, then the tetrahedron is regular.
(b) Is a tetrahedron necessarily regular if five dihedral angles are congruent?
1988 AMC 12/AHSME, 23
The six edges of a tetrahedron $ABCD$ measure $7$, $13$, $18$, $27$, $36$ and $41$ units. If the length of edge $AB$ is $41$, then the length of edge $CD$ is
$ \textbf{(A)}\ 7\qquad\textbf{(B)}\ 13\qquad\textbf{(C)}\ 18\qquad\textbf{(D)}\ 27\qquad\textbf{(E)}\ 36 $
1970 IMO Longlists, 45
Let $M$ be an interior point of tetrahedron $V ABC$. Denote by $A_1,B_1, C_1$ the points of intersection of lines $MA,MB,MC$ with the planes $VBC,V CA,V AB$, and by $A_2,B_2, C_2$ the points of intersection of lines $V A_1, VB_1, V C_1$ with the sides $BC,CA,AB$.
[b](a)[/b] Prove that the volume of the tetrahedron $V A_2B_2C_2$ does not exceed one-fourth of the volume of $V ABC$.
[b](b)[/b] Calculate the volume of the tetrahedron $V_1A_1B_1C_1$ as a function of the volume of $V ABC$, where $V_1$ is the point of intersection of the line $VM$ with the plane $ABC$, and $M$ is the barycenter of $V ABC$.
2013 Princeton University Math Competition, 7
A tetrahedron $ABCD$ satisfies $AB=6$, $CD=8$, and $BC=DA=5$. Let $V$ be the maximum value of $ABCD$ possible. If we can write $V^4=2^n3^m$ for some integers $m$ and $n$, find $mn$.
1976 IMO Longlists, 2
Let $P$ be a set of $n$ points and $S$ a set of $l$ segments. It is known that:
$(i)$ No four points of $P$ are coplanar.
$(ii)$ Any segment from $S$ has its endpoints at $P$.
$(iii)$ There is a point, say $g$, in $P$ that is the endpoint of a maximal number of segments from $S$ and that is not a vertex of a tetrahedron having all its edges in $S$.
Prove that $l \leq \frac{n^2}{3}$
1967 IMO Longlists, 40
Prove that a tetrahedron with just one edge length greater than $1$ has volume at most $ \frac{1}{8}.$
2004 Germany Team Selection Test, 1
Let $a_{ij}$ $i=1,2,3$; $j=1,2,3$ be real numbers such that $a_{ij}$ is positive for $i=j$ and negative for $i\neq j$.
Prove the existence of positive real numbers $c_{1}$, $c_{2}$, $c_{3}$ such that the numbers \[a_{11}c_{1}+a_{12}c_{2}+a_{13}c_{3},\qquad a_{21}c_{1}+a_{22}c_{2}+a_{23}c_{3},\qquad a_{31}c_{1}+a_{32}c_{2}+a_{33}c_{3}\] are either all negative, all positive, or all zero.
[i]Proposed by Kiran Kedlaya, USA[/i]
2011 National Olympiad First Round, 33
What is the largest volume of a sphere which touches to a unit sphere internally and touches externally to a regular tetrahedron whose corners are over the unit sphere?
$\textbf{(A)}\ \frac13 \qquad\textbf{(B)}\ \frac14 \qquad\textbf{(C)}\ \frac12\left ( 1 - \frac1{\sqrt3} \right ) \qquad\textbf{(D)}\ \frac12\left ( \frac{2\sqrt2}{\sqrt3} - 1 \right ) \qquad\textbf{(E)}\ \text{None}$
1997 Estonia National Olympiad, 3
A sphere is inscribed in a regular tetrahedron. Another regular tetrahedron is inscribed in the sphere. Find the ratio of the volumes of these two tetrahedra.
1953 Poland - Second Round, 5
Calculate the volume $ V $ of tetrahedron $ ABCD $ given the length $ d $ of edge $ AB $ and the area $ S $ of the projection of the tetrahedron on the plane perpendicular to the line $ AB $.
1984 AIME Problems, 9
In tetrahedron $ABCD$, edge $AB$ has length 3 cm. The area of face $ABC$ is 15 $\text{cm}^2$ and the area of face $ABD$ is 12 $\text{cm}^2$. These two faces meet each other at a $30^\circ$ angle. Find the volume of the tetrahedron in $\text{cm}^3$.
2006 AMC 10, 24
Centers of adjacent faces of a unit cube are joined to form a regular octahedron. What is the volume of this octahedron?
$ \textbf{(A) } \frac 18 \qquad \textbf{(B) } \frac 16 \qquad \textbf{(C) } \frac 14 \qquad \textbf{(D) } \frac 13 \qquad \textbf{(E) } \frac 12$
2017 Simon Marais Mathematical Competition, B1
Maryam labels each vertex of a tetrahedron with the sum of the lengths of the three edges meeting at that vertex.
She then observes that the labels at the four vertices of the tetrahedron are all equal. For each vertex of the tetrahedron, prove that the lengths of the three edges meeting at that vertex are the three side lengths of a triangle.
1997 Taiwan National Olympiad, 5
Let $ABCD$ is a tetrahedron. Show that
a)If $AB=CD,AC=DB,AD=BC$ then triangles $ABC,ABD,ACD,BCD$ are acute.
b)If the triangles $ABC,ABD,ACD,BCD$ have the same area , then $AB=CD,AC=DB,AD=BC$.