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 Today's Calculation Of Integral, 759
Tags: calculus , integration , 3d geometry , tetrahedron , geometric transformation , rotation , geometry
Given a regular tetrahedron $PQRS$ with side length $d$. Find the volume of the solid generated by a rotation around the line passing through $P$ and the midpoint $M$ of $QR$.

Estonia Open Senior - geometry, 1995.1.3
Tags: geometry , 3d geometry , tetrahedron , area of a triangle , area
We call a tetrahedron a "trirectangular " if it has a vertex (we call this is called a "right-angled" vertex) in which the planes of the three sides of the tetrahedron intersect at right angles. Prove the "three-dimensional Pythagorean theorem": The square of the area of the opposite face of the "right-angled" vertex of the ""trirectangular " tetrahedron is equal to the sum of the squares of the areas of three other sides of the tetrahedron .

2023 CCA Math Bonanza, TB2
Tags: geometry , 3d geometry , tetrahedron
How many ways are there to color a tetrahedron’s faces, edges, and vertices in red, green, and blue so that no face shares a color with any of its edges, and no edge shares a color with any of its endpoints? (Rotations and reflections are considered distinct.) [i]Tiebreaker #2[/i]

1986 IMO Longlists, 11
Tags: geometry , 3d geometry , tetrahedron , trigonometry , congruent triangles
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.

1952 Poland - Second Round, 6
Tags: geometry , 3d geometry , tetrahedron
Prove that a plane that passes: a) through the centers of two opposite edges of the tetrahedron and b) through the center of one of the other edges of the tetrahedron divides the tetrahedron into two parts of equal volumes. Will the thesis remain true if we reject assumption (b) ?

2011 Tuymaada Olympiad, 4
Tags: 3d geometry , number theory , geometry
In a set of consecutive positive integers, there are exactly $100$ perfect cubes and $10$ perfect fourth powers. Prove that there are at least $2000$ perfect squares in the set.

1991 Arnold's Trivium, 86
Tags: geometry , 3d geometry , tetrahedron , icosahedron
Through the centre of a cube (tetrahedron, icosahedron) draw a straight line in such a way that the sum of the squares of its distances from the vertices is a) minimal, b) maximal.

2014 Miklós Schweitzer, 10
Tags: 3d geometry , sphere , geometry
To each vertex of a given triangulation of the two-dimensional sphere, we assign a convex subset of the plane. Assume that the three convex sets corresponding to the three vertices of any two-dimensional face of the triangulation have at least one point in common. Show that there exist four vertices such that the corresponding convex sets have at least one point in common.

1979 IMO Longlists, 68
Tags: trigonometry , geometry , 3d geometry , maximization
We consider a point $P$ in a plane $p$ and a point $Q \not\in p$. Determine all the points $R$ from $p$ for which \[ \frac{QP+PR}{QR} \] is maximum.

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

1983 USAMO, 4
Tags: 3d geometry , tetrahedron , 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.

1974 IMO Longlists, 44
Tags: 3d geometry , sphere , geometry
We are given $n$ mass points of equal mass in space. We define a sequence of points $O_1,O_2,O_3,\ldots $ as follows: $O_1$ is an arbitrary point (within the unit distance of at least one of the $n$ points); $O_2$ is the centre of gravity of all the $n$ given points that are inside the unit sphere centred at $O_1$;$O_3$ is the centre of gravity of all of the $n$ given points that are inside the unit sphere centred at $O_2$; etc. Prove that starting from some $m$, all points $O_m,O_{m+1},O_{m+2},\ldots$ coincide.

2009 AMC 12/AHSME, 17
Tags: geometry , 3d geometry , probability
Each face of a cube is given a single narrow stripe painted from the center of one edge to the center of its opposite edge. The choice of the edge pairing is made at random and independently for each face. What is the probability that there is a continuous stripe encircling the cube? $ \textbf{(A)}\ \frac {1}{8}\qquad \textbf{(B)}\ \frac {3}{16}\qquad \textbf{(C)}\ \frac {1}{4} \qquad \textbf{(D)}\ \frac {3}{8}\qquad \textbf{(E)}\ \frac {1}{2}$

2023 AMC 12/AHSME, 13
Tags: 3d geometry , diagonal
A rectangular box $\mathcal{P}$ has distinct edge lengths $a, b,$ and $c$. The sum of the lengths of all $12$ edges of $\mathcal{P}$ is $13$, the sum of the areas of all $6$ faces of $\mathcal{P}$ is $\frac{11}{2}$, and the volume of $\mathcal{P}$ is $\frac{1}{2}$. What is the length of the longest interior diagonal connecting two vertices of $\mathcal{P}$? $\textbf{(A)}~2\qquad\textbf{(B)}~\frac{3}{8}\qquad\textbf{(C)}~\frac{9}{8}\qquad\textbf{(D)}~\frac{9}{4}\qquad\textbf{(E)}~\frac{3}{2}$

2017 Adygea Teachers' Geometry Olympiad, 4
Tags: geometry , 3d geometry , tetrahedron , volume
A regular tetrahedron $SABC$ of volume $V$ is given. The midpoints $D$ and $E$ are taken on $SA$ and $SB$ respectively and the point $F$ is taken on the edge $SC$ such that $SF: FC = 1: 3$. Find the volume of the pentahedron $FDEABC$.

1975 Bundeswettbewerb Mathematik, 2
Tags: combinatorial geometry , polyhedron , 3d geometry , edge , geometry
Prove that in each polyhedron there exist two faces with the same number of edges.

1981 Romania Team Selection Tests, 2.
Tags: geometry , 3d geometry , tetrahedron
Consider a tetrahedron $OABC$ with $ABC$ equilateral. Let $S$ be the area of the triangle of sides $OA$, $OB$ and $OC$. Show that $V\leqslant \dfrac12 RS$ where $R$ is the circumradius and $V$ is the volume of the tetrahedron. [i]Stere Ianuș[/i]

1990 Tournament Of Towns, (255) 3
Tags: geometry , 3d geometry , icosahedron , dodecahedron
(a) Some vertices of a dodecahedron are to be marked so that each face contains a marked vertex. What is the smallest number of marked vertices for which this is possible? (b) Answer the same question, but for an icosahedron. (G. Galperin, Moscow) (Recall that a dodecahedron has $12$ pentagonal faces which meet in threes at each vertex, while an icosahedron has $20$ triangular faces which meet in fives at each vertex.)

1953 Miklós Schweitzer, 7
Tags: 3d geometry
[b]7.[/b] Consider four real numbers $t_{1},t_{2},t_{3},t_{4}$ such that each is less than the sum of the others. Show that there exists a tetrahedron whose faces have areas $t_{1},t_{2}, t_{3}$ and $t_{4},$ respectively. [b](G. 9)[/b]

2013 AMC 10, 14
Tags: geometry , 3d geometry , asymptote
A solid cube of side length $1$ is removed from each corner of a solid cube of side length $3$. How many edges does the remaining solid have? $\textbf{(A) }36\qquad \textbf{(B) }60\qquad \textbf{(C) }72\qquad \textbf{(D) }84\qquad \textbf{(E) }108\qquad$

1996 Czech and Slovak Match, 3
Tags: inequalities , solid geometry , pyramid , geometry , 3d geometry
The base of a regular quadrilateral pyramid $\pi$ is a square with side length $2a$ and its lateral edge has length a$\sqrt{17}$. Let $M$ be a point inside the pyramid. Consider the five pyramids which are similar to $\pi$ , whose top vertex is at $M$ and whose bases lie in the planes of the faces of $\pi$ . Show that the sum of the surface areas of these five pyramids is greater or equal to one fifth the surface of $\pi$ , and find for which $M$ equality holds.

2014 BAMO, 1
Tags: cube , geometry , 3d geometry , counting , combinatorics , arrangement
The four bottom corners of a cube are colored red, green, blue, and purple. How many ways are there to color the top four corners of the cube so that every face has four different colored corners? Prove that your answer is correct.

1953 Moscow Mathematical Olympiad, 255
Tags: geometry , 3d geometry , pyramid , cube
Divide a cube into three equal pyramids.

2015 BMT Spring, Tie 1
Tags: geometry , 3d geometry
Compute the surface area of a rectangular prism with side lengths $2, 3, 4$.

2001 National High School Mathematics League, 9
Tags: 3d geometry , geometry
The length of edge of cube $ABCD-A_1B_1C_1D_1$ is $1$, then the distance between lines $A_1C_1$ and $BD_1$ is________.