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

2008 HMNT, 10
Tags: geometry , 3D geometry
Find the largest positive integer $n$ such that $n^3 + 4n^2 - 15n - 18$ is the cube of an integer.

1973 AMC 12/AHSME, 2
Tags: geometry , 3D geometry
One thousand unit cubes are fastened together to form a large cube with edge length 10 units; this is painted and then separated into the original cubes. The number of these unit cubes which have at least one face painted is $ \textbf{(A)}\ 600 \qquad \textbf{(B)}\ 520 \qquad \textbf{(C)}\ 488 \qquad \textbf{(D)}\ 480 \qquad \textbf{(E)}\ 400$

2004 Postal Coaching, 17
Tags: geometry , 3D geometry , number theory unsolved , number theory
In a system of numeration with base $B$ , there are $n$ one-digit numbers less than $B$ whose cubes have $B-1$ in the units-digits place. Determine the relation between $n$ and $B$

1981 Miklós Schweitzer, 5
Tags: geometry , 3D geometry , vector , advanced fields , advanced fields unsolved
Let $ K$ be a convex cone in the $ n$-dimensional real vector space $ \mathbb{R}^n$, and consider the sets $ A\equal{}K \cup (\minus{}K)$ and $ B\equal{}(\mathbb{R}^n \setminus A) \cup \{ 0 \}$ ($ 0$ is the origin). Show that one can find two subspaces in $ \mathbb{R}^n$ such that together they span $ \mathbb{R}^n$, and one of them lies in $ A$ and the other lies in $ B$. [i]J. Szucs[/i]

2005 AIME Problems, 9
Tags: AMC , AIME , geometry , 3D geometry , probability , rotation
Twenty seven unit cubes are painted orange on a set of four faces so that two non-painted faces share an edge. The $27$ cubes are randomly arranged to form a $3\times 3 \times 3$ cube. Given the probability of the entire surface area of the larger cube is orange is $\frac{p^a}{q^br^c},$ where $p$,$q$, and $r$ are distinct primes and $a$,$b$, and $c$ are positive integers, find $a+b+c+p+q+r$.

1973 IMO Shortlist, 7
Tags: geometry , 3D geometry , tetrahedron , IMO Shortlist
Given a tetrahedron $ABCD$, let $x = AB \cdot CD$, $y = AC \cdot BD$, and $z = AD \cdot BC$. Prove that there exists a triangle with edges $x, y, z.$

2006 Putnam, B1
Tags: Putnam , quadratics , geometry , 3D geometry , function , Stanford , college
Show that the curve $x^{3}+3xy+y^{3}=1$ contains only one set of three distinct points, $A,B,$ and $C,$ which are the vertices of an equilateral triangle.

2021 Harvard-MIT Mathematics Tournament., 5
Tags: 3D geometry , combinatorics
A convex polyhedron has $n$ faces that are all congruent triangles with angles $36^{\circ}, 72^{\circ}$, and $72^{\circ}$. Determine, with proof, the maximum possible value of $n$.

2000 Czech And Slovak Olympiad IIIA, 5
Tags: 3D geometry , tetrahedron , geometry , Volume
Monika made a paper model of a tetrahedron whose base is a right-angled triangle. When she cut the model along the legs of the base and the median of a lateral face corresponding to one of the legs, she obtained a square of side a. Compute the volume of the tetrahedron.

2010 Purple Comet Problems, 16
Tags: geometry , 3D geometry
Half the volume of a 12 foot high cone-shaped pile is grade A ore while the other half is grade B ore. The pile is worth \$62. One-third of the volume of a similarly shaped 18 foot pile is grade A ore while the other two-thirds is grade B ore. The second pile is worth \$162. Two-thirds of the volume of a similarly shaped 24 foot pile is grade A ore while the other one-third is grade B ore. What is the value in dollars (\$) of the 24 foot pile?

2002 District Olympiad, 3
Tags: geometry , 3D geometry , pyramid , polyhedron , angles
Consider the regular pyramid $VABCD$ with the vertex in $V$ which measures the angle formed by two opposite lateral edges is $45^o$. The points $M,N,P$ are respectively, the projections of the point $A$ on the line $VC$, the symmetric of the point $M$ with respect to the plane $(VBD)$ and the symmetric of the point $N$ with respect to $O$. ($O$ is the center of the base of the pyramid.) a) Show that the polyhedron $MDNBP$ is a regular pyramid. b) Determine the measure of the angle between the line $ND$ and the plane $(ABC) $

2008 AMC 12/AHSME, 11
Tags: geometry , 3D geometry , AMC
A cone-shaped mountain has its base on the ocean floor and has a height of $ 8000$ feet. The top $ \frac{1}{8}$ of the volume of the mountain is above water. What is the depth of the ocean at the base of the mountain, in feet? $ \textbf{(A)}\ 4000 \qquad \textbf{(B)}\ 2000(4\minus{}\sqrt{2}) \qquad \textbf{(C)}\ 6000 \qquad \textbf{(D)}\ 6400 \qquad \textbf{(E)}\ 7000$

1964 All Russian Mathematical Olympiad, 053
Tags: geometry , 3D geometry , tetrahedron , minimum
We have to divide a cube onto $k$ non-overlapping tetrahedrons. For what smallest $k$ is it possible?

2023 LMT Fall, 22
Tags: geometry , 3D geometry , algebra
Consider all pairs of points $(a,b,c)$ and $(d,e, f )$ in the $3$-D coordinate system with $ad +be +c f = -2023$. What is the least positive integer that can be the distance between such a pair of points? [i]Proposed by William Hua[/i]

1973 IMO Longlists, 7
Tags: geometry , 3D geometry , tetrahedron
Given a tetrahedron $ABCD$. Let $x = AB \cdot CD, y = AC \cdot BD$ and $z = AD\cdot BC$. Prove that there exists a triangle with the side lengths $x, y$ and $z$.

2001 IMC, 3
Tags: geometry , 3D geometry , sphere
Find the maximum number of points on a sphere of radius $1$ in $\mathbb{R}^n$ such that the distance between any two of these points is strictly greater than $\sqrt{2}$.

2006 MOP Homework, 4
Tags: geometry , 3D geometry , tetrahedron , Pythagorean Theorem , geometry unsolved
Let $ABCD$ be a tetrahedron and let $H_{a},H_{b},H_{c},H_{d}$ be the orthocenters of triangles $BCD,CDA,DAB,ABC$, respectively. Prove that lines $AH_{a},BH_{b},CH_{c}, DH_{d}$ are concurrent if and only if $AB^2 + CD^2 = AC^2 + BD^2 = AD^2 + BC^2$

2008 Princeton University Math Competition, A10
Tags: geometry , trigonometry , 3D geometry , sphere , combinatorial geometry
A cuboctahedron is the convex hull of (smallest convex set containing) the $12$ points $(\pm 1, \pm 1, 0), (\pm 1, 0, \pm 1), (0, \pm 1, \pm 1)$. Find the cosine of the solid angle of one of the triangular faces, as viewed from the origin. (Take a figure and consider the set of points on the unit sphere centered on the origin such that the ray from the origin through the point intersects the fi gure. The area of that set is the solid angle of the fi gure as viewed from the origin.)

2003 Tournament Of Towns, 1
Tags: geometry , 3D geometry , pyramid , sphere , geometry unsolved
A triangular pyramid $ABCD$ is given. Prove that $\frac Rr > \frac ah$, where $R$ is the radius of the circumscribed sphere, $r$ is the radius of the inscribed sphere, $a$ is the length of the longest edge, $h$ is the length of the shortest altitude (from a vertex to the opposite face).

2010 Contests, 3
Tags: geometry , 3D geometry , trigonometry , vector , linear algebra , matrix , inequalities
What is the biggest shadow that a cube of side length $1$ can have, with the sun at its peak? Note: "The biggest shadow of a figure with the sun at its peak" is understood to be the biggest possible area of the orthogonal projection of the figure on a plane.

2005 AMC 12/AHSME, 22
Tags: geometry , 3D geometry , sphere
A rectangular box $ P$ is inscribed in a sphere of radius $ r$. The surface area of $ P$ is 384, and the sum of the lengths of its 12 edges is 112. What is $ r$? $ \textbf{(A)}\ 8 \qquad \textbf{(B)}\ 10 \qquad \textbf{(C)}\ 12 \qquad \textbf{(D)}\ 14 \qquad \textbf{(E)}\ 16$

2012 AMC 12/AHSME, 22
Tags: geometry , 3D geometry , tetrahedron , prism , AMC
Distinct planes $p_1,p_2,....,p_k$ intersect the interior of a cube $Q$. Let $S$ be the union of the faces of $Q$ and let $ P =\bigcup_{j=1}^{k}p_{j} $. The intersection of $P$ and $S$ consists of the union of all segments joining the midpoints of every pair of edges belonging to the same face of $Q$. What is the difference between the maximum and minimum possible values of $k$? $ \textbf{(A)}\ 8\qquad\textbf{(B)}\ 12\qquad\textbf{(C)}\ 20\qquad\textbf{(D)}\ 23\qquad\textbf{(E)}\ 24 $

1992 National High School Mathematics League, 14
Tags: geometry , 3D geometry , geometric transformation , reflection
$l,m$ are skew lines. Three points $A,B,C$ on line $l$ satisfy that $AB=BC$. Projection of $A,B,C$ on $m$ are $D,E,F$. If $|AD|=\sqrt{15},|BE|=\frac{7}{2}|CF|=\sqrt{10}$, find the distance between $l$ and $m$.

1984 IMO Longlists, 11
Tags: geometry , 3D geometry , tetrahedron , Volume , geometric inequality , IMO Shortlist
Prove that the volume of a tetrahedron inscribed in a right circular cylinder of volume $1$ does not exceed $\frac{2}{3 \pi}.$

2008 AMC 12/AHSME, 8
Tags: geometry , 3D geometry , AMC
What is the volume of a cube whose surface area is twice that of a cube with volume $ 1$? $ \textbf{(A)}\ \sqrt{2} \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 2\sqrt{2} \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 8$