Olympiad problems

1997 Polish MO Finals, 3

In a tetrahedron $ABCD$, the medians of the faces $ABD$, $ACD$, $BCD$ from $D$ make equal angles with the corresponding edges $AB$, $AC$, $BC$. Prove that each of these faces has area less than or equal to the sum of the areas of the other two faces. [hide="Comment"][i]Equivalent version of the problem:[/i] $ABCD$ is a tetrahedron. $DE$, $DF$, $DG$ are medians of triangles $DBC$, $DCA$, $DAB$. The angles between $DE$ and $BC$, between $DF$ and $CA$, and between $DG$ and $AB$ are equal. Show that: area $DBC$ $\leq$ area $DCA$ + area $DAB$. [/hide]

1973 IMO Shortlist, 9

Tags: tetrahedron , 3d geometry , minimization , perimeter , geometry

Let $Ox, Oy, Oz$ be three rays, and $G$ a point inside the trihedron $Oxyz$. Consider all planes passing through $G$ and cutting $Ox, Oy, Oz$ at points $A,B,C$, respectively. How is the plane to be placed in order to yield a tetrahedron $OABC$ with minimal perimeter ?

2000 Bundeswettbewerb Mathematik, 3

Tags: tetrahedron , geometry , 3d geometry , sphere

For each vertex of a given tetrahedron, a sphere passing through that vertex and the midpoints of the edges outgoing from this vertex is constructed. Prove that these four spheres pass through a single point.

1974 Polish MO Finals, 1

Tags: tetrahedron , 3d geometry , geometry

In a tetrahedron $ABCD$ the edges $AB$ and $CD$ are perpendicular and $\angle ACB =\angle ADB$. Prove that the plane through $AB$ and the midpoint of the edge $CD$, is perpendicular to $CD$.

1973 USAMO, 1

Tags: 3d geometry , tetrahedron , geometry

Two points $ P$ and $ Q$ lie in the interior of a regular tetrahedron $ ABCD$. Prove that angle $ PAQ < 60^\circ$.

1966 IMO Longlists, 23

Tags: tetrahedron , geometry , 3d geometry

Three faces of a tetrahedron are right triangles, while the fourth is not an obtuse triangle. [i](a) [/i]Prove that a necessary and sufficient condition for the fourth face to be a right triangle is that at some vertex exactly two angles are right. [i](b)[/i] Prove that if all the faces are right triangles, then the volume of the tetrahedron equals one -sixth the product of the three smallest edges not belonging to the same face.

2008 District Olympiad, 1

Tags: tetrahedron , rhombus , geometry , square , plane , 3d geometry

A regular tetrahedron is sectioned with a plane after a rhombus. Prove that the rhombus is square.

Ukrainian TYM Qualifying - geometry, 2013.17

Tags: geometry , concurrency , tetrahedron , 3d geometry

Through the point of intersection of the medians of each of the faces a tetrahedron is drawn perpendicular to this face. Prove that all these four lines intersect at one point if and only if the four lines containing the heights of this tetrahedron intersect at one point .

2006 Kazakhstan National Olympiad, 6

Tags: tetrahedron , perpendicular , geometry , 3d geometry , plane

In the tetrahedron $ ABCD $ from the vertex $ A $, the perpendiculars $ AB '$, $ AC' $ are drawn, $ AD '$ on planes dividing dihedral angles at edges $ CD $, $ BD $, $ BC $ in half. Prove that the plane $ (B'C'D ') $ is parallel to the plane $ (BCD) $.

1991 Turkey Team Selection Test, 3

Tags: geometry , 3d geometry , tetrahedron , pyramid

Let $U$ be the sum of lengths of sides of a tetrahedron (triangular pyramid) with vertices $O,A,B,C$. Let $V$ be the volume of the convex shape whose vertices are the midpoints of the sides of the tetrahedron. Show that $V\leq \frac{(U-|OA|-|BC| )(U-|OB|-|AC| )(U-|OC|-|AB| )}{(2^{7} \cdot 3)}$.

1997 Taiwan National Olympiad, 5

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

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 $AB=CD,AC=DB,AD=BC$.

1986 Austrian-Polish Competition, 6

Tags: tetrahedron , geometry , concentric , inscribed , circumscribed , 3d geometry , sphere

Let $M$ be the set of all tetrahedra whose inscribed and circumscribed spheres are concentric. If the radii of these spheres are denoted by $r$ and $R$ respectively, find the possible values of $R/r$ over all tetrahedra from $M$ .

1983 USAMO, 4

Tags: tetrahedron , geometry , 3d geometry

Six segments $S_1, S_2, S_3, S_4, S_5,$ and $S_6$ are given in a plane. These are congruent to the edges $AB, AC, AD, BC, BD,$ and $CD$, respectively, of a tetrahedron $ABCD$. Show how to construct a segment congruent to the altitude of the tetrahedron from vertex $A$ with straight-edge and compasses.

2002 AMC 12/AHSME, 24

Tags: geometry , tetrahedron , 3d geometry , prism

Let $ABCD$ be a regular tetrahedron and let $E$ be a point inside the face $ABC$. Denote by $s$ the sum of the distances from $E$ to the faces $DAB$, $DBC$, $DCA$, and by $S$ the sum of the distances from $E$ to the edges $AB$, $BC$, $CA$. Then $\dfrac sS$ equals $\textbf{(A) }\sqrt2\qquad\textbf{(B) }\dfrac{2\sqrt2}3\qquad\textbf{(C) }\dfrac{\sqrt6}2\qquad\textbf{(D) }2\qquad\textbf{(E) }3$

1988 IMO Longlists, 37

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

[b]i.)[/b] Four balls of radius 1 are mutually tangent, three resting on the floor and the fourth resting on the others. A tedrahedron, each of whose edges has length $ s,$ is circumscribed around the balls. Find the value of $ s.$ [b]ii.)[/b] Suppose that $ ABCD$ and $ EFGH$ are opposite faces of a retangular solid, with $ \angle DHC \equal{} 45^{\circ}$ and $ \angle FHB \equal{} 60^{\circ}.$ Find the cosine of $ \angle BHD.$

KoMaL A Problems 2022/2023, A.837

Tags: geometry , inequalities , linear algebra , geometric inequality , 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]

1969 IMO Longlists, 55

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

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 $1$.

2005 Sharygin Geometry Olympiad, 22

Tags: tetrahedron , concurrency , geometry , 3d geometry , centroid

Perpendiculars at their centers of gravity (points of intersection of medians) are restored to the faces of the tetrahedron. Prove that the projections of the three perpendiculars to the fourth face intersect at one point.

2019 Jozsef Wildt International Math Competition, W. 60

Tags: inequalities , tetrahedron , inradius , exradius , 3d geometry

In all tetrahedron $ABCD$ holds [list=1] [*] $(n(n+2))^{\frac{1}{n}} \sum \limits_{cyc} \left(\frac{(h_a-r)^2}{(h_a^n-r^n)(h_a^{n+2}-r^{n+2})}\right)^{\frac{1}{n}}\leq \frac{1}{r^2}$ [*] $(n(n+2))^{\frac{1}{n}} \sum \limits_{cyc} \left(\frac{(r_a-r)^2}{(r_a^n-r^n)(r_a^{n+2}-r^{n+2})}\right)^{\frac{1}{n}}\leq \frac{1}{r^2}$ [/list] for all $n\in \mathbb{N}^*$

1980 Vietnam National Olympiad, 1

Tags: geometry , ratio , tetrahedron , 3d geometry

Prove that for any tetrahedron in space, it is possible to find two perpendicular planes such that ratio between the projections of the tetrahedron on the two planes lies in the interval $[\frac{1}{\sqrt{2}}, \sqrt{2}].$

1980 IMO Longlists, 15

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

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

2007 AMC 12/AHSME, 16

Tags: symmetry , geometry , rotation , counting , tetrahedron , 3d geometry , distinguishability

Each face of a regular tetrahedron is painted either red, white or blue. Two colorings are considered indistinguishable if two congruent tetrahedra with those colorings can be rotated so that their appearances are identical. How many distinguishable colorings are possible? $ \textbf{(A)}\ 15 \qquad \textbf{(B)}\ 18 \qquad \textbf{(C)}\ 27 \qquad \textbf{(D)}\ 54 \qquad \textbf{(E)}\ 81$

1968 IMO, 4

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

Prove that every tetrahedron has a vertex whose three edges have the right lengths to form a triangle.

2002 IMC, 10

Tags: tetrahedron , inequalities , 3d geometry

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 $\gamma > \beta \cos\sigma + \alpha \cos\rho$.

2001 Romania National Olympiad, 2

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

In the tetrahedron $OABC$ we denote by $\alpha,\beta,\gamma$ the measures of the angles $\angle BOC,\angle COA,$ and $\angle AOB$, respectively. Prove the inequality \[\cos^2\alpha+\cos^2\beta+\cos^2\gamma<1+2\cos\alpha\cos\beta\cos\gamma \]