Olympiad problems

2014 Contests, 3

A tetrahedron $ABCD$ with acute-angled faces is inscribed in a sphere with center $O$. A line passing through $O$ perpendicular to plane $ABC$ crosses the sphere at point $D'$ that lies on the opposide side of plane $ABC$ than point $D$. Line $DD'$ crosses plane $ABC$ in point $P$ that lies inside the triangle $ABC$. Prove, that if $\angle APB=2\angle ACB$, then $\angle ADD'=\angle BDD'$.

1977 IMO Longlists, 56

Tags: 3d geometry , tetrahedron , circumcircle , geometry

The four circumcircles of the four faces of a tetrahedron have equal radii. Prove that the four faces of the tetrahedron are congruent triangles.

2020 Sharygin Geometry Olympiad, 24

Tags: incenter , 3d geometry , tetrahedron , geometry

Let $I$ be the incenter of a tetrahedron $ABCD$, and $J$ be the center of the exsphere touching the face $BCD$ containing three remaining faces (outside these faces). The segment $IJ$ meets the circumsphere of the tetrahedron at point $K$. Which of two segments $IJ$ and $JK$ is longer?

1975 Vietnam National Olympiad, 3

Tags: 3d geometry , tetrahedron , geometry

Let $ABCD$ be a tetrahedron with $BA \perp AC,DB \perp (BAC)$. Denote by $O$ the midpoint of $AB$, and $K$ the foot of the perpendicular from $O$ to $DC$. Suppose that $AC = BD$. Prove that $\frac{V_{KOAC}}{V_{KOBD}}=\frac{AC}{BD}$ if and only if $2AC \cdot BD = AB^2$.

1984 IMO Shortlist, 13

Tags: geometry , 3d geometry , tetrahedron , volume , geometric inequality

Prove that the volume of a tetrahedron inscribed in a right circular cylinder of volume $1$ does not exceed $\frac{2}{3 \pi}.$

1982 Polish MO Finals, 6

Tags: geometry , 3d geometry , tetrahedron , dihedral angle

Prove that the sum of dihedral angles in an arbitrary tetrahedron is greater than $2\pi$

1949-56 Chisinau City MO, 60

Tags: geometry , distance , regular tetrahedron , tetrahedron , 3d geometry

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.

1994 All-Russian Olympiad, 7

Tags: altitude , geometry , 3d geometry , tetrahedron , sphere

The altitudes $AA_1,BB_1,CC_1,DD_1$ of a tetrahedron $ABCD$ intersect in the center $H$ of the sphere inscribed in the tetrahedron $A_1B_1C_1D_1$. Prove that the tetrahedron $ABCD$ is regular. (D. Tereshin)

1992 AMC 12/AHSME, 19

Tags: geometry , 3d geometry , tetrahedron , ratio

For each vertex of a solid cube, consider the tetrahedron determined by the vertex and the midpoints of the three edges that meet at that vertex. The portion of the cube that remains when these eight tetrahedra are cut away is called a [i]cuboctahedron[/i]. The ratio of the volume of the cuboctahedron to the volume of the original cube is closest to which of these? $ \textbf{(A)}\ 75\%\qquad\textbf{(B)}\ 78\%\qquad\textbf{(C)}\ 81\%\qquad\textbf{(D)}\ 84\%\qquad\textbf{(E)}\ 87\% $

1988 Polish MO Finals, 3

Tags: 3d geometry , tetrahedron , sphere , trigonometry , geometry

Find the largest possible volume for a tetrahedron which lies inside a hemisphere of radius $1$.

2001 All-Russian Olympiad Regional Round, 11.6

Tags: geometry , tetrahedron , 3d geometry , concurrency

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.

1986 IMO Longlists, 30

Tags: inequalities , geometry , 3d geometry , icosahedron , tetrahedron , octahedron , dodecahedron

Prove that a convex polyhedron all of whose faces are equilateral triangles has at most $30$ edges.

2019 CMIMC, 4

Tags: geometry , 3d geometry , tetrahedron

Suppose $\mathcal{T}=A_0A_1A_2A_3$ is a tetrahedron with $\angle A_1A_0A_3 = \angle A_2A_0A_1 = \angle A_3A_0A_2 = 90^\circ$, $A_0A_1=5, A_0A_2=12$ and $A_0A_3=9$. A cube $A_0B_0C_0D_0E_0F_0G_0H_0$ with side length $s$ is inscribed inside $\mathcal{T}$ with $B_0\in \overline{A_0A_1}, D_0 \in \overline{A_0A_2}, E_0 \in \overline{A_0A_3}$, and $G_0\in \triangle A_1A_2A_3$; what is $s$?

1980 IMO Shortlist, 15

Tags: geometry , 3d geometry , tetrahedron , sphere , triangle inequality , angle

Prove that the sum of the six angles subtended at an interior point of a tetrahedron by its six edges is greater than 540°.

2011 Mediterranean Mathematics Olympiad, 3

Tags: geometry , 3d geometry , tetrahedron , sphere , trigonometry , integration

A regular tetrahedron of height $h$ has a tetrahedron of height $xh$ cut off by a plane parallel to the base. When the remaining frustrum is placed on one of its slant faces on a horizontal plane, it is just on the point of falling over. (In other words, when the remaining frustrum is placed on one of its slant faces on a horizontal plane, the projection of the center of gravity G of the frustrum is a point of the minor base of this slant face.) Show that $x$ is a root of the equation $x^3 + x^2 + x = 2$.

2013 Saint Petersburg Mathematical Olympiad, 3

Tags: 3d geometry , tetrahedron , parallelogram , geometry

Let $M$ and $N$ are midpoint of edges $AB$ and $CD$ of the tetrahedron $ABCD$, $AN=DM$ and $CM=BN$. Prove that $AC=BD$. S. Berlov

KoMaL A Problems 2022/2023, A.837

Tags: geometry , linear algebra , geometric inequality , inequalities , 3d geometry , tetrahedron

Let all the edges of tetrahedron $A_1A_2A_3A_4$ be tangent to sphere $S$. Let $\displaystyle a_i$ denote the length of the tangent from $A_i$ to $S$. Prove that \[\bigg(\sum_{i=1}^4 \frac 1{a_i}\bigg)^{\!\!2}> 2\bigg(\sum_{i=1}^4 \frac1{a_i^2}\bigg). \] [i]Submitted by Viktor Vígh, Szeged[/i]

1982 Tournament Of Towns, (030) 4

Tags: geometry , 3d geometry , tetrahedron , vector , regular polygon , regular tetrahedron

(a) $K_1,K_2,..., K_n$ are the feet of the perpendiculars from an arbitrary point $M$ inside a given regular $n$-gon to its sides (or sides produced). Prove that the sum $\overrightarrow{MK_1} + \overrightarrow{MK_2} + ... + \overrightarrow{MK_n}$ equals $\frac{n}{2}\overrightarrow{MO}$, where $O$ is the centre of the $n$-gon. (b) Prove that the sum of the vectors whose origin is an arbitrary point $M$ inside a given regular tetrahedron and whose endpoints are the feet of the perpendiculars from $M$ to the faces of the tetrahedron equals $\frac43 \overrightarrow{MO}$, where $O$ is the centre of the tetrahedron. (VV Prasolov, Moscow)

1969 IMO Longlists, 12

Tags: geometry , 3d geometry , tetrahedron , locus , locus problems

$(CZS 1)$ Given a unit cube, find the locus of the centroids of all tetrahedra whose vertices lie on the sides of the cube.

2000 IMC, 4

Tags: 3d geometry , tetrahedron , inequalities

Let $OABC$ be a tetrahedon with $\angle BOC=\alpha,\angle COA =\beta$ and $\angle AOB =\gamma$. The angle between the faces $OAB$ and $OAC$ is $\sigma$ and the angle between the faces $OAB$ and $OBC$ is $\rho$. Show that $\gamma > \beta \cos\sigma + \alpha \cos\rho$.

1971 IMO Shortlist, 7

Tags: geometry , 3d geometry , tetrahedron , trigonometry , distance

All faces of the tetrahedron $ABCD$ are acute-angled. Take a point $X$ in the interior of the segment $AB$, and similarly $Y$ in $BC, Z$ in $CD$ and $T$ in $AD$. [b]a.)[/b] If $\angle DAB+\angle BCD\ne\angle CDA+\angle ABC$, then prove none of the closed paths $XYZTX$ has minimal length; [b]b.)[/b] If $\angle DAB+\angle BCD=\angle CDA+\angle ABC$, then there are infinitely many shortest paths $XYZTX$, each with length $2AC\sin k$, where $2k=\angle BAC+\angle CAD+\angle DAB$.

2015 Math Prize for Girls Olympiad, 2

Tags: geometry , 3d geometry , tetrahedron

A tetrahedron $T$ is inside a cube $C$. Prove that the volume of $T$ is at most one-third the volume of $C$.

1988 Irish Math Olympiad, 1

Tags: geometry , 3d geometry , pyramid , tetrahedron

A pyramid with a square base, and all its edges of length $2$, is joined to a regular tetrahedron, whose edges are also of length $2$, by gluing together two of the triangular faces. Find the sum of the lengths of the edges of the resulting solid.

1990 All Soviet Union Mathematical Olympiad, 532

Tags: inequalities , geometric inequality , 3d geometry , tetrahedron , distance , altitude

If every altitude of a tetrahedron is at least $1$, show that the shortest distance between each pair of opposite edges is more than $2$.

2008 Romania National Olympiad, 1

Tags: 3d geometry , tetrahedron , pyramid , geometry

A tetrahedron has the side lengths positive integers, such that the product of any two opposite sides equals 6. Prove that the tetrahedron is a regular triangular pyramid in which the lateral sides form an angle of at least 30 degrees with the base plane.