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: 2265

2011 National Olympiad First Round, 33
Tags: geometry , 3D geometry , sphere , tetrahedron
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? $\textbf{(A)}\ \frac13 \qquad\textbf{(B)}\ \frac14 \qquad\textbf{(C)}\ \frac12\left ( 1 - \frac1{\sqrt3} \right ) \qquad\textbf{(D)}\ \frac12\left ( \frac{2\sqrt2}{\sqrt3} - 1 \right ) \qquad\textbf{(E)}\ \text{None}$

2010 Tournament Of Towns, 3
Tags: geometry , 3D geometry , octahedron , geometry unsolved
Is it possible to cover the surface of a regular octahedron by several regular hexagons without gaps and overlaps? (A regular octahedron has $6$ vertices, each face is an equilateral triangle, each vertex belongs to $4$ faces.)

1998 Romania National Olympiad, 4
Tags: geometry , tetrahedron , trigonometry , 3D geometry
Let $A_1A_2...A_n$ be a regular polygon ($n > 4$), $T$ be the common point of $A_1A_2$ and $A_{n-1}A_n$ and $M$ be a point in the interior of the triangle $A_1A_nT$. Show that the equality $$\sum_{i=1}^{n-1} \frac{\sin^2 \left(\angle A_iMA_{i+1}\right)}{d(M,A_iA_{i+1}}=\frac{\sin^2 \left(\angle A_1MA_n\right)}{d(M,A_1A_n} $$ holds if and only if $M$ belongs to the circumcircle of the polygon.

2015 Finnish National High School Mathematics Comp, 2
Tags: 3D geometry , pyramid , Volume , geometry
The lateral edges of a right square pyramid are of length $a$. Let $ABCD$ be the base of the pyramid, $E$ its top vertex and $F$ the midpoint of $CE$. Assuming that $BDF$ is an equilateral triangle, compute the volume of the pyramid.

2015 Romania National Olympiad, 3
Tags: pyramid , angle , 3D geometry , geometry
Let $VABC$ be a regular triangular pyramid with base $ABC$, of center $O$. Points $I$ and $H$ are the center of the inscribed circle, respectively the orthocenter $\vartriangle VBC$. Knowing that $AH = 3 OI$, determine the measure of the angle between the lateral edge of the pyramid and the plane of the base.

2008 AIME Problems, 5
Tags: geometry , 3D geometry , AMC , AIME , search , geometric transformation , rotation
A right circular cone has base radius $ r$ and height $ h$. The cone lies on its side on a flat table. As the cone rolls on the surface of the table without slipping, the point where the cone's base meets the table traces a circular arc centered at the point where the vertex touches the table. The cone first returns to its original position on the table after making $ 17$ complete rotations. The value of $ h/r$ can be written in the form $ m\sqrt {n}$, where $ m$ and $ n$ are positive integers and $ n$ is not divisible by the square of any prime. Find $ m \plus{} n$.

1985 IMO Longlists, 87
Tags: geometry , 3D geometry , sphere , tetrahedron , function , IMO Shortlist
Determine the radius of a sphere $S$ that passes through the centroids of each face of a given tetrahedron $T$ inscribed in a unit sphere with center $O$. Also, determine the distance from $O$ to the center of $S$ as a function of the edges of $T.$

1964 Kurschak Competition, 1
Tags: ratio , geometry , tetrahedron , 3D geometry
$ABC$ is an equilateral triangle. $D$ and$ D'$ are points on opposite sides of the plane $ABC$ such that the two tetrahedra $ABCD$ and $ABCD'$ are congruent (but not necessarily with the vertices in that order). If the polyhedron with the five vertices $A, B, C, D, D'$ is such that the angle between any two adjacent faces is the same, find $DD'/AB$ .

2009 AMC 12/AHSME, 5
Tags: geometry , 3D geometry , AMC
One dimension of a cube is increased by $ 1$, another is decreased by $ 1$, and the third is left unchanged. The volume of the new rectangular solid is $ 5$ less than that of the cube. What was the volume of the cube? $ \textbf{(A)}\ 8 \qquad \textbf{(B)}\ 27 \qquad \textbf{(C)}\ 64 \qquad \textbf{(D)}\ 125 \qquad \textbf{(E)}\ 216$

2020 Sharygin Geometry Olympiad, 24
Tags: geometry , incenter , 3D geometry , tetrahedron , Sharygin 2020
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?

1974 Bulgaria National Olympiad, Problem 6
Tags: inequalities , geometry , 3D geometry , geometrical inequalities , pyramid , Geometric Inequalities
In triangle pyramid $MABC$ at least two of the plane angles next to the edge $M$ are not equal to each other. Prove that if the bisectors of these angles form the same angle with the angle bisector of the third plane angle, the following inequality is true $$8a_1b_1c_1\le a^2a_1+b^2b_1+c^2c_1$$ where $a,b,c$ are sides of triangle $ABC$ and $a_1,b_1,c_1$ are edges crossed respectively with $a,b,c$. [i]V. Petkov[/i]

1986 IMO Shortlist, 20
Tags: geometry , 3D geometry , tetrahedron , trigonometry , congruent triangles , IMO Shortlist
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.

2012-2013 SDML (Middle School), 11
Tags: geometry , 3D geometry , NEEDS DIAGRAM , SDML Middle School , 2012-13 SDML Round 2
Six different-sized cubes are glued together, one on top of the other. The bottom cube has edge length $8$. Each of the other cubes has four vertices at the midpoints of the edges of the cube below it as shown. The entire solid is then dipped in red paint. What is the total area of the red-painted surface on the solid? (will insert image here later) $\text{(A) }630\qquad\text{(B) }632\qquad\text{(C) }648\qquad\text{(D) }694\qquad\text{(E) }756$

1985 Tournament Of Towns, (096) 5
Tags: geometry , rectangle , combinatorics , combinatorial geometry , projection , parallelepiped , 3D geometry
A square is divided into rectangles. A "chain" is a subset $K$ of the set of these rectangles such that there exists a side of the square which is covered by projections of rectangles of $K$ and such that no point of this side is a projection of two inner points of two inner points of two different rectangles of $K$. (a) Prove that every two rectangles in such a division are members of a certain "chain". (b) Solve the similar problem for a cube, divided into rectangular parallelopipeds (in the definition of chain , replace "side" by"edge") . (A.I . Golberg, V.A. Gurevich)

1977 IMO Longlists, 1
Tags: geometry , 3D geometry , pyramid , geometry proposed
A pentagon $ABCDE$ inscribed in a circle for which $BC<CD$ and $AB<DE$ is the base of a pyramid with vertex $S$. If $AS$ is the longest edge starting from $S$, prove that $BS>CS$.

2021 Saint Petersburg Mathematical Olympiad, 3
Tags: geometry , 3D geometry , pyramid , circumcircle
In the pyramid $SA_1A_2 \cdots A_n$, all sides are equal. Let point $X_i$ be the midpoint of arc $A_iA_{i+1}$ in the circumcircle of $\triangle SA_iA_{i+1}$ for $1 \le i \le n$ with indices taken mod $n$. Prove that the circumcircles of $X_1A_2X_2, X_2A_3X_3, \cdots, X_nA_1X_1$ have a common point.

2009 Sharygin Geometry Olympiad, 23
Tags: geometry , 3D geometry , prism , geometry proposed
Is it true that for each $ n$, the regular $ 2n$-gon is a projection of some polyhedron having not greater than $ n \plus{} 2$ faces?

1947 Moscow Mathematical Olympiad, 138
Tags: geometry , combinatorial geometry , 3D geometry , points in space
In space, $n$ wire triangles are situated so that any two of them have a common vertex and each vertex is the vertex of $k$ triangles. Find all $n$ and $k$ for which this is possible.

2013 Stanford Mathematics Tournament, 5
Tags: geometry , 3D geometry , prism
A polygonal prism is made from a flexible material such that the two bases are regular $2^n$-gons $(n>1)$ of the same size. The prism is bent to join the two bases together without twisting, giving a figure with $2^n$ faces. The prism is then repeatedly twisted so that each edge of one base becomes aligned with each edge of the base exactly once. For an arbitrary $n$, what is the sum of the number of faces over all of these configurations (including the non-twisted case)?

2004 National High School Mathematics League, 9
Tags: geometry , 3D geometry
In cube $ABCD-A_1B_1C_1D_1$, the degree of dihedral angle $A-BD_1-A_1$ is________.

1970 Czech and Slovak Olympiad III A, 2
Tags: geometry , 3D geometry , tetrahedron , right triangle , right angle , similar , similar triangles
Determine whether there is a tetrahedron $ABCD$ with the longest edge of length 1 such that all its faces are similar right triangles with right angles at vertices $B,C.$ If so, determine which edge is the longest, which is the shortest and what is its length.

1992 USAMO, 4
Tags: geometry , 3D geometry , sphere , Hi
Chords $AA^{\prime}$, $BB^{\prime}$, $CC^{\prime}$ of a sphere meet at an interior point $P$ but are not contained in a plane. The sphere through $A$, $B$, $C$, $P$ is tangent to the sphere through $A^{\prime}$, $B^{\prime}$, $C^{\prime}$, $P$. Prove that $\, AA' = BB' = CC'$.

1967 Poland - Second Round, 6
Tags: geometry , Concyclic , sphere , 3D geometry
Prove that the points $ A_1, A_2, \ldots, A_n $ ($ n \geq 7 $) located on the surface of the sphere lie on a circle if and only if the planes tangent to the surface of the sphere at these points have a common point or are parallel to one straight line.

1951 Moscow Mathematical Olympiad, 195
Tags: pyramid , concentric circles , tangential , geometry , 3D geometry , polygon
We have two concentric circles. A polygon is circumscribed around the smaller circle and is contained entirely inside the greater circle. Perpendiculars from the common center of the circles to the sides of the polygon are extended till they intersect the greater circle. Each of the points obtained is connected with the endpoints of the corresponding side of the polygon . When is the resulting star-shaped polygon the unfolding of a pyramid?

1967 IMO Longlists, 25
Tags: geometry , 3D geometry , sphere , angles , IMO Shortlist , IMO Longlist
Three disks of diameter $d$ are touching a sphere in their centers. Besides, every disk touches the other two disks. How to choose the radius $R$ of the sphere in order that axis of the whole figure has an angle of $60^\circ$ with the line connecting the center of the sphere with the point of the disks which is at the largest distance from the axis ? (The axis of the figure is the line having the property that rotation of the figure of $120^\circ$ around that line brings the figure in the initial position. Disks are all on one side of the plane, passing through the center of the sphere and orthogonal to the axis).