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

1966 IMO Shortlist, 21
Tags: geometry , 3d geometry , area , volume , geometric inequality
Prove that the volume $V$ and the lateral area $S$ of a right circular cone satisfy the inequality \[\left( \frac{6V}{\pi}\right)^2 \leq \left( \frac{2S}{\pi \sqrt 3}\right)^3\] When does equality occur?

2005 Harvard-MIT Mathematics Tournament, 5
Tags: geometry , 3d geometry , sphere , rectangle , pythagorean theorem
A cube with side length $2$ is inscribed in a sphere. A second cube, with faces parallel to the first, is inscribed between the sphere and one face of the first cube. What is the length of a side of the smaller cube?

2004 District Olympiad, 3
Tags: geometry , 3d geometry , tetrahedron , geometric transformation , rotation
On the tetrahedron $ ABCD $ make the notation $ M,N,P,Q, $ for the midpoints of $ AB,CD,AC, $ respectively, $ BD. $ Additionally, we know that $ MN $ is the common perpendicular of $ AB,CD, $ and $ PQ $ is the common perpendicular of $ AC,BD. $ Show that $ AB=CD, BC=DA, AC=BD. $

2008 AMC 10, 3
Tags: geometry , 3d geometry
Assume that $ x$ is a positive real number. Which is equivalent to $ \sqrt[3]{x\sqrt{x}}$? $ \textbf{(A)}\ x^{1/6} \qquad \textbf{(B)}\ x^{1/4} \qquad \textbf{(C)}\ x^{3/8} \qquad \textbf{(D)}\ x^{1/2} \qquad \textbf{(E)}\ x$

1989 Swedish Mathematical Competition, 3
Tags: perfect square , number theory , geometry , 3d geometry
Find all positive integers $n$ such that $n^3 - 18n^2 + 115n - 391$ is the cube of a positive intege

2015 AMC 12/AHSME, 16
Tags: geometry , 3d geometry , tetrahedron , analytic geometry , calculus , vector
Tetrahedron $ABCD$ has $AB=5$, $AC=3$, $BC=4$, $BD=4$, $AD=3$, and $CD=\tfrac{12}5\sqrt2$. What is the volume of the tetrahedron? $\textbf{(A) }3\sqrt2\qquad\textbf{(B) }2\sqrt5\qquad\textbf{(C) }\dfrac{24}5\qquad\textbf{(D) }3\sqrt3\qquad\textbf{(E) }\dfrac{24}5\sqrt2$

1983 IMO Longlists, 71
Tags: geometry , 3d geometry , sphere , trigonometry , combinatorics , partition
Prove that every partition of $3$-dimensional space into three disjoint subsets has the following property: One of these subsets contains all possible distances; i.e., for every $a \in \mathbb R^+$, there are points $M$ and $N$ inside that subset such that distance between $M$ and $N$ is exactly $a.$

1978 Romania Team Selection Test, 6
Tags: geometry , 3d geometry
Show that there is no polyhedron whose projection on the plane is a nondegenerate triangle.

1998 Flanders Math Olympiad, 2
Tags: geometry , 3d geometry , analytic geometry , vector , trigonometry
Given a cube with edges of length 1, $e$ the midpoint of $[bc]$, and $m$ midpoint of the face $cdc_1d_1$, as on the figure. Find the area of intersection of the cube with the plane through the points $a,m,e$. [img]http://www.mathlinks.ro/Forum/album_pic.php?pic_id=279[/img]

1965 IMO, 3
Tags: geometry , 3d geometry , tetrahedron , prism
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.

1974 IMO Longlists, 17
Tags: 3d geometry , sphere , geometry
Show that there exists a set $S$ of $15$ distinct circles on the surface of a sphere, all having the same radius and such that $5$ touch exactly $5$ others, $5$ touch exactly $4$ others, and $5$ touch exactly $3$ others. [i][General Problem: http://www.artofproblemsolving.com/Forum/viewtopic.php?f=46&t=384764][/i]

1993 Irish Math Olympiad, 5
Tags: rectangle , 3d geometry , geometry
$ (a)$ The rectangle $ PQRS$ with $ PQ\equal{}l$ and $ QR\equal{}m$ $ (l,m \in \mathbb{N})$ is divided into $ lm$ unit squares. Prove that the diagonal $ PR$ intersects exactly $ l\plus{}m\minus{}d$ of these squares, where $ d\equal{}(l,m)$. $ (b)$ A box with edge lengths $ l,m,n \in \mathbb{N}$ is divided into $ lmn$ unit cubes. How many of the cubes does a main diagonal of the box intersect?

1990 AMC 12/AHSME, 10
Tags: geometry , 3d geometry
An $11\times 11\times 11$ wooden cube is formed by gluing together $11^3$ unit cubes. What is the greatest number of unit cubes that can be seen from a single point? $\textbf{(A) }328\qquad \textbf{(B) }329\qquad \textbf{(C) }330\qquad \textbf{(D) }331\qquad \textbf{(E) }332\qquad$

1999 Baltic Way, 2
Tags: geometry , 3d geometry , number theory
Determine all positive integers $n$ with the property that the third root of $n$ is obtained by removing its last three decimal digits.

2005 Purple Comet Problems, 2
Tags: geometry , 3d geometry
We glue together $990$ one inch cubes into a $9$ by $10$ by $11$ inch rectangular solid. Then we paint the outside of the solid. How many of the original $990$ cubes have just one of their sides painted?

1982 Tournament Of Towns, (028) 2
Tags: polyhedron , 3d geometry , combinatorial geometry , geometry
Does there exist a polyhedron (not necessarily convex) which could have the following complete list of edges? $AB, AC, BC, BD, CD, DE, EF, EG, FG, FH, GH, AH$. [img]http://1.bp.blogspot.com/-wTdNfQHG5RU/XVk1Bf4wpqI/AAAAAAAAKhA/8kc6u9KqOgg_p1CXim2LZ1ANFXFiWgnYACK4BGAYYCw/s1600/TOT%2B1982%2BAutum%2BS2.png[/img]

2013 ELMO Shortlist, 2
Tags: algebra , polynomial , 3d geometry , modular arithmetic , number theory , geometry
For what polynomials $P(n)$ with integer coefficients can a positive integer be assigned to every lattice point in $\mathbb{R}^3$ so that for every integer $n \ge 1$, the sum of the $n^3$ integers assigned to any $n \times n \times n$ grid of lattice points is divisible by $P(n)$? [i]Proposed by Andre Arslan[/i]

1968 Putnam, A4
Tags: geometry , 3d geometry , sphere
Let $S^{2}\subset \mathbb{R}^{3}$ be the unit sphere. Show that for any $n$ points on $ S^{2}$, the sum of the squares of the $\frac{n(n-1)}{2}$ distances between them is at most $n^{2}$.

2024 Czech-Polish-Slovak Junior Match, 2
Tags: product , perfect cube , geometry , 3d geometry , number theory
How many non-empty subsets of $\{1,2,\dots,11\}$ are there with the property that the product of its elements is the cube of an integer?

VMEO III 2006 Shortlist, G3
Tags: geometry , 3d geometry , tetrahedron , geometric inequality
The tetrahedron $OABC$ has all angles at vertex $O$ equal to $60^o$. Prove that $$AB \cdot BC + BC \cdot CA + CA \cdot AB \ge OA^2 + OB^2 + OC^2$$

2012 German National Olympiad, 5
Tags: geometry , inradius , 3d geometry , tetrahedron , inequalities , geometric inequality
Let $a,b$ be the lengths of two nonadjacent edges of a tetrahedron with inradius $r$. Prove that \[r<\frac{ab}{2(a+b)}.\]

1956 Polish MO Finals, 6
Tags: 3d geometry , geometry , sphere
Given a sphere of radius $ R $ and a plane $ \alpha $ having no common points with this sphere. A point $ S $ moves in the plane $ \alpha $, which is the vertex of a cone tangent to the sphere along a circle with center $ C $. Find the locus of point $ C $. [hide=another is Polish MO 1967 p6] [url=https://artofproblemsolving.com/community/c6h3388032p31769739]here[/url][/hide]

2009 Sharygin Geometry Olympiad, 23
Tags: 3d geometry , prism , geometry
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?

1991 Baltic Way, 18
Tags: geometry , 3d geometry , sphere
Is it possible to place two non-intersecting tetrahedra of volume $\frac{1}{2}$ into a sphere with radius $1$?

1992 Bundeswettbewerb Mathematik, 3
Tags: geometry , 3d geometry , sphere
Given is a triangle $ABC$ with side lengths $a, b,c$. Three spheres touch each other in pairs and also touch the plane of the triangle at points $A,B$ and $C$, respectively. Determine the radii of these spheres.