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

2010 Stanford Mathematics Tournament, 8
Tags: 3d geometry , sphere , tetrahedron , geometry
A sphere of radius $1$ is internally tangent to all four faces of a regular tetrahedron. Find the tetrahedron's volume.

1982 Czech and Slovak Olympiad III A, 1
Tags: 3d geometry , geometry , collinear , tetrahedron
Given a tetrahedron $ABCD$ and inside the tetrahedron points $K, L, M, N$ that do not lie on a plane. Denote also the centroids of $P$, $Q$, $R$, $S$ of the tetrahedrons $KBCD$, $ALCD$, $ABMD$, $ABCN$ do not lie on a plane. Let $T$ be the centroid of the tetrahedron ABCD, $T_o$ be the centroid of the tetrahedron $PQRS$ and $T_1$ be the centroid of the tetrahedron $KLMN$. a) Prove that the points $T, T_0, T_1$ lie in one straight line. b) Determine the ratio $|T_0T| : |T_0 T_1|$.

1965 IMO Shortlist, 3
Tags: 3d geometry , geometry , prism , tetrahedron
Given the tetrahedron $ABCD$ whose edges $AB$ and $CD$ have lengths $a$ and $b$ respectively. The distance between the skew lines $AB$ and $CD$ is $d$, and the angle between them is $\omega$. Tetrahedron $ABCD$ is divided into two solids by plane $\epsilon$, parallel to lines $AB$ and $CD$. The ratio of the distances of $\epsilon$ from $AB$ and $CD$ is equal to $k$. Compute the ratio of the volumes of the two solids obtained.

1992 Putnam, A6
Tags: 3d geometry , symmetry , sphere , probability , tetrahedron , geometric transformation , geometry
Four points are chosen at random on the surface of a sphere. What is the probability that the center of the sphere lies inside the tetrahedron whose vertices are at the four points?

2005 Sharygin Geometry Olympiad, 11.6
Tags: 3d geometry , concurrency , geometry , sphere , tetrahedron
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.

1979 Austrian-Polish Competition, 5
Tags: 3d geometry , circumcenter , congruent , tetrahedron , geometry , incenter
The circumcenter and incenter of a given tetrahedron coincide. Prove that all its faces are congruent.

2003 AIME Problems, 4
Tags: 3d geometry , number theory , relatively prime , tetrahedron , vector , geometry , ratio
In a regular tetrahedron the centers of the four faces are the vertices of a smaller tetrahedron. The ratio of the volume of the smaller tetrahedron to that of the larger is $m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m+n.$

1990 Bundeswettbewerb Mathematik, 4
Tags: midpoint , 3d geometry , sphere , tetrahedron , geometry
Suppose that every two opposite edges of a tetrahedron are orthogonal. Show that the midpoints of the six edges lie on a sphere.

1969 IMO, 3
Tags: 3d geometry , triangle inequality , tetrahedron , geometry
For each of $k=1,2,3,4,5$ find necessary and sufficient conditions on $a>0$ such that there exists a tetrahedron with $k$ edges length $a$ and the remainder length $1$.

1995 Brazil National Olympiad, 4
Tags: 3d geometry , rectangle , geometry , tetrahedron , circumcircle
A regular tetrahedron has side $L$. What is the smallest $x$ such that the tetrahedron can be passed through a loop of twine of length $x$?

1991 IMTS, 5
Tags: 3d geometry , area of a triangle , heron's formula , algebra , tetrahedron , rotation , geometry
The sides of $\triangle ABC$ measure 11,20, and 21 units. We fold it along $PQ,QR,RP$ where $P,Q,R$ are the midpoints of its sides until $A,B,C$ coincide. What is the volume of the resulting tetrahedron?

2010 All-Russian Olympiad Regional Round, 11.6
Tags: 3d geometry , volume , geometry , tetrahedron
At the base of the quadrangular pyramid $SABCD$ lies the parallelogram $ABCD$. Prove that for any point $O$ inside the pyramid, the sum of the volumes of the tetrahedra $OSAB$ and $OSCD$ is equal to the sum of the volumes of the tetrahedra $OSBC$ and $OSDA$ .

2019 Jozsef Wildt International Math Competition, W. 68
Tags: tetrahedron , 3d geometry , exradius , inradius , inequalities
In all tetrahedron $ABCD$ holds [list=1] [*] $\displaystyle{\sum \limits_{cyc}\frac{h_a-r}{h_a+r}\geq \sum \limits_{cyc}\frac{h_a^t-r^t}{(h_a+r)^t}}$ [*] $\displaystyle{\sum \limits_{cyc}\frac{2r_a-r}{2r_a+r}\geq \sum \limits_{cyc}\frac{2r_a^t-r^t}{(2r_a+r)^t}}$ [/list] for all $t\in [0,1]$

1966 IMO Shortlist, 60
Tags: geometry , 3d geometry , tetrahedron , sphere
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.

2013 Stanford Mathematics Tournament, 9
Tags: 3d geometry , geometry , tetrahedron , incenter
In tetrahedron $ABCD$, $AB=4$, $CD=7$, and $AC=AD=BC=BD=5$. Let $I_A$, $I_B$, $I_C$, and $I_D$ denote the incenters of the faces opposite vertices $A$, $B$, $C$, and $D$, respectively. It is provable that $AI_A$ intersects $BI_B$ at a point $X$, and $CI_C$ intersects $DI_D$ at a point $Y$. Compute $XY$.

2007 All-Russian Olympiad, 7
Tags: 3d geometry , inequalities , geometry , tetrahedron
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]

2006 AMC 10, 24
Tags: 3d geometry , pyramid , octahedron , tetrahedron , geometry
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 Iranian Geometry Olympiad, 5
Tags: geometry , 3d geometry , sphere , tetrahedron
Sphere $S$ touches a plane. Let $A,B,C,D$ be four points on the plane such that no three of them are collinear. Consider the point $A'$ such that $S$ in tangent to the faces of tetrahedron $A'BCD$. Points $B',C',D'$ are defined similarly. Prove that $A',B',C',D'$ are coplanar and the plane $A'B'C'D'$ touches $S$. [i]Proposed by Alexey Zaslavsky (Russia)[/i]

2015 Sharygin Geometry Olympiad, P24
Tags: 3d geometry , 3-dimensional geometry , sphere , tetrahedron , geometry
The insphere of a tetrahedron ABCD with center $O$ touches its faces at points $A_1,B_1,C_1$ and $D_1$. a) Let $P_a$ be a point such that its reflections in lines $OB,OC$ and $OD$ lie on plane $BCD$. Points $P_b, P_c$ and $P_d$ are defined similarly. Prove that lines $A_1P_a,B_1P_b,C_1P_c$ and $D_1P_d$ concur at some point $ P$. b) Let $I$ be the incenter of $A_1B_1C_1D_1$ and $A_2$ the common point of line $A_1I $ with plane $B_1C_1D_1$. Points $B_2, C_2, D_2$ are defined similarly. Prove that $P$ lies inside $A_2B_2C_2D_2$.

2008 Sharygin Geometry Olympiad, 24
Tags: 3d geometry , geometry , tetrahedron , inequalities
(I.Bogdanov, 11) Let $ h$ be the least altitude of a tetrahedron, and $ d$ the least distance between its opposite edges. For what values of $ t$ the inequality $ d>th$ is possible?

1995 Czech And Slovak Olympiad IIIA, 1
Tags: 3d geometry , geometric inequalities , geometry , angle , tetrahedron
Suppose that tetrahedron $ABCD$ satisfies $\angle BAC+\angle CAD+\angle DAB = \angle ABC+\angle CBD+\angle DBA = 180^o$. Prove that $CD \ge AB$.

1999 Croatia National Olympiad, Problem 1
Tags: 3d geometry , tetrahedron , geometry
For every edge of a tetrahedron, we consider a plane through its midpoint that is perpendicular to the opposite edge. Prove that these six planes intersect in a point symmetric to the circumcenter of the tetrahedron with respect to its centroid.

1996 Vietnam National Olympiad, 2
Tags: symmetry , 3d geometry , tetrahedron , sphere , angle bisector , circumcircle , geometry
Given a trihedral angle Sxyz. A plane (P) not through S cuts Sx,Sy,Sz respectively at A,B,C. On the plane (P), outside triangle ABC, construct triangles DAB,EBC,FCA which are confruent to the triangles SAB,SBC,SCA respectively. Let (T) be the sphere lying inside Sxyz, but not inside the tetrahedron SABC, toucheing the planes containing the faces of SABC. Prove that (T) touches the plane (P) at the circumcenter of triangle DEF.

1964 IMO Shortlist, 6
Tags: 3d geometry , tetrahedron , geometry , analytic geometry
In tetrahedron $ABCD$, vertex $D$ is connected with $D_0$, the centrod if $\triangle ABC$. Line parallel to $DD_0$ are drawn through $A,B$ and $C$. These lines intersect the planes $BCD, CAD$ and $ABD$ in points $A_2, B_1,$ and $C_1$, respectively. Prove that the volume of $ABCD$ is one third the volume of $A_1B_1C_1D_0$. Is the result if point $D_o$ is selected anywhere within $\triangle ABC$?

1989 IMO Longlists, 67
Tags: angle , tetrahedron , 3d geometry , geometric inequality , geometry
Prove that the intersection of a plane and a regular tetrahedron can be an obtuse-angled triangle and that the obtuse angle in any such triangle is always smaller than $ 120^{\circ}.$