This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 619

1997 Bundeswettbewerb Mathematik, 1
Tags: tetrahedron , coloring , combinatorics
Three faces of a regular tetrahedron are painted in white and the remaining one in black. Initially, the tetrahedron is positioned on a plane with the black face down. It is then tilted several times over its edges. After a while it returns to its original position. Can it now have a white face down?

1976 IMO Longlists, 51
Tags: combinatorics , 3d geometry , tetrahedron , geometry
Four swallows are catching a fly. At first, the swallows are at the four vertices of a tetrahedron, and the fly is in its interior. Their maximal speeds are equal. Prove that the swallows can catch the fly.

2015 AoPS Mathematical Olympiad, 2
Tags: 3d geometry , tetrahedron , sphere , geometry
In tetrahedron $ABCD$, let $V$ be the volume of the tetrahedron and $R$ the radius of the sphere that it tangent to all four faces of the tetrahedron. Let $P$ be the surface area of the tetrahedron. Prove that $$r=\frac{3V}{P}.$$ [i]Proposed by CaptainFlint.[/i]

1984 Tournament Of Towns, (061) O2
Tags: tetrahedron , geometry , 3d geometry , plane , altitude , parallel
Six altitudes are constructed from the three vertices of the base of a tetrahedron to the opposite sides of the three lateral faces. Prove that all three straight lines joining two base points of the altitudes in each lateral face are parallel to a certain plane. (IF Sharygin, Moscow)

1982 Polish MO Finals, 6
Tags: 3d geometry , dihedral angle , tetrahedron , geometry
Prove that the sum of dihedral angles in an arbitrary tetrahedron is greater than $2\pi$

1966 IMO Longlists, 60
Tags: 3d geometry , sphere , tetrahedron , geometry
Prove that the sum of the distances of the vertices of a regular tetrahedron from the center of its circumscribed sphere is less than the sum of the distances of these vertices from any other point in space.

1998 Tuymaada Olympiad, 4
Tags: plane , dissecting plane , 3d geometry , solid geometry , geometry , tetrahedron
Given the tetrahedron $ABCD$, whose opposite edges are equal, that is, $AB=CD, AC=BD$ and $BC=AD$. Prove that exist exactly $6$ planes intersecting the triangular angles of the tetrahedron and dividing the total surface and volume of this tetrahedron in half.

1964 German National Olympiad, 3
Tags: geometry , area , sphere , 3d geometry , tetrahedron
Given a (not necessarily regular) tetrahedron, all of its sides are equal in area. Prove that the following points then coincide: a) the center of the inscribed sphere, i.e. all four side surfaces internally touching sphere, b) the center of the surrounding sphere, i.e. the sphere passing through the four vertixes.

1984 AIME Problems, 9
Tags: geometry , 3d geometry , tetrahedron , trigonometry
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 $\text{cm}^3$.

1974 Chisinau City MO, 84
Tags: 3d geometry , similar , tetrahedron , geometry
a) Let $S$ and $P$ be the area and perimeter of some triangle. The straight lines on which its sides are located move to the outside by a distance $h$. What will be the area and perimeter of the triangle formed by the three obtained lines? b) Let $V$ and $S$ be the volume and surface area of some tetrahedron. The planes on which its faces are located are moved to the outside by a distance $h$. What will be the volume and surface area of the tetrahedron formed by the three obtained planes?

2014 Sharygin Geometry Olympiad, 7
Tags: tetrahedron , 3d geometry , geometry
Prove that the smallest dihedral angle between faces of an arbitrary tetrahedron is not greater than the dihedral angle between faces of a regular tetrahedron. (S. Shosman, O. Ogievetsky)

1990 French Mathematical Olympiad, Problem 4
Tags: 3d geometry , tetrahedron , geometry
(a) What is the maximum area of a triangle with vertices in a given square (or on its boundary)? (b) What is the maximum volume of a tetrahedron with vertices in a given cube (or on its boundary)?

2003 Tournament Of Towns, 6
Tags: tetrahedron , 3d geometry , inradius , parallelogram , geometry
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.

1992 National High School Mathematics League, 3
Tags: tetrahedron , 3d geometry , geometry
Areas of four surfaces of a tetrahedron are $S_1,S_2,S_3,S_4$. And the largest one of them is $S$. $\lambda=\frac{S_1+S_2+S_3+S_4}{S}$, then $\lambda$ always satisfies $\text{(A)}2<\lambda\leq4\qquad\text{(B)}3<\lambda<4\qquad\text{(C)}2.5<\lambda\leq4.5\qquad\text{(D)}3.5<\lambda<5.5$

2005 Tuymaada Olympiad, 4
Tags: trigonometry , triangle inequality , tetrahedron , 3d geometry , geometry , inequalities
In a triangle $ABC$, let $A_{1}$, $B_{1}$, $C_{1}$ be the points where the excircles touch the sides $BC$, $CA$ and $AB$ respectively. Prove that $A A_{1}$, $B B_{1}$ and $C C_{1}$ are the sidelenghts of a triangle. [i]Proposed by L. Emelyanov[/i]

2001 AIME Problems, 12
Tags: tetrahedron , 3d geometry , number theory , geometry , relatively prime
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 $m+n$.

Kvant 2023, M2747
Tags: tetrahedron , geometry , 3d geometry
In the tetrahedron $ABCD,$ on the continuation of the edges $AB, AC$ and $AD$, three points were marked for point $A{},$ located from $A{}$ at a distance equal to the semi-perimeter of the triangle $BCD.$ Similarly, we marked three points corresponding to vertices $B, C$ and $D.$ Prove that if there is a sphere touching all the edges of the tetrahedron $ABCD$, then the marked 12 points lie on the same sphere. [i]Proposed by V. Alexandrov[/i]

1998 All-Russian Olympiad Regional Round, 11.7
Tags: tetrahedron , 3d geometry , volume , geometry
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 $\Phi$.

2007 Bundeswettbewerb Mathematik, 3
Tags: 3d geometry , tetrahedron , geometry , vector
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))?$

MathLinks Contest 3rd, 3
Tags: tetrahedron , geometry , 3d geometry
We say that a tetrahedron is [i]median [/i] if and only if for each vertex the plane that passes through the midpoints of the edges emerging from the vertex is tangent to the inscribed sphere. Also a tetrahedron is called [i]regular [/i] if all its faces are congruent. Prove that a tetrahedron is regular if and only if it is median.

2014 Bulgaria National Olympiad, 3
Tags: geometry , combinatorics , tetrahedron , 3d geometry , sphere
A real number $f(X)\neq 0$ is assigned to each point $X$ in the space. It is known that for any tetrahedron $ABCD$ with $O$ the center of the inscribed sphere, we have : \[ f(O)=f(A)f(B)f(C)f(D). \] Prove that $f(X)=1$ for all points $X$. [i]Proposed by Aleksandar Ivanov[/i]

1986 IMO Longlists, 11
Tags: congruent triangles , geometry , trigonometry , tetrahedron , 3d geometry
Prove that the sum of the face angles at each vertex of a tetrahedron is a straight angle if and only if the faces are congruent triangles.

2008 All-Russian Olympiad, 4
Tags: function , tetrahedron , circumcircle , 3d geometry , geometry , analytic geometry , sphere
Each face of a tetrahedron can be placed in a circle of radius $ 1$. Show that the tetrahedron can be placed in a sphere of radius $ \frac{3}{2\sqrt2}$.

2001 All-Russian Olympiad Regional Round, 11.6
Tags: concurrency , geometry , tetrahedron , 3d geometry
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.

1949-56 Chisinau City MO, 60
Tags: regular tetrahedron , tetrahedron , 3d geometry , geometry , distance
Show that the sum of the distances from any point of a regular tetrahedron to its faces is equal to the height of this tetrahedron.