Found problems: 2265
1996 ITAMO, 3
Given a cube of unit side. Let $A$ and $B$ be two opposite vertex. Determine the radius of the sphere, with center inside the cube, tangent to the three faces of the cube with common point $A$ and tangent to the three sides with common point $B$.
2014 NIMO Problems, 2
How many $2 \times 2 \times 2$ cubes must be added to a $8 \times 8 \times 8$ cube to form a $12 \times 12 \times 12$ cube?
[i]Proposed by Evan Chen[/i]
2022 CCA Math Bonanza, T3
The smallest possible volume of a cylinder that will fit nine spheres of radius 1 can be expressed as $x\pi$ for some value of $x$. Compute $x$.
[i]2022 CCA Math Bonanza Team Round #3[/i]
2023 LMT Fall, 7
Isabella is making sushi. She slices a piece of salmon into the shape of a solid triangular prism. The prism is $2$ cm thick, and its triangular faces have side lengths $7$ cm, $ 24$cm, and $25$ cm. Find the volume of this piece of salmon in cm$^3$.
[i]Proposed by Isabella Li[/i]
2012 CHMMC Spring, 2
A convex octahedron in Cartesian space contains the origin in its interior. Two of its vertices are on the $x$-axis, two are on the $y$-axis, and two are on the $z$-axis. One triangular face $F$ has side lengths $\sqrt{17}$, $\sqrt{37}$, $\sqrt{52}$. A second triangular face $F_0$ has side lengths $\sqrt{13}$, $\sqrt{29}$, $\sqrt{34}$. What is the minimum possible volume of the octahedron?
1958 AMC 12/AHSME, 8
Which of these four numbers $ \sqrt{\pi^2},\,\sqrt[3]{.8},\,\sqrt[4]{.00016},\,\sqrt[3]{\minus{}1}\cdot \sqrt{(.09)^{\minus{}1}}$, is (are) rational:
$ \textbf{(A)}\ \text{none}\qquad
\textbf{(B)}\ \text{all}\qquad
\textbf{(C)}\ \text{the first and fourth}\qquad
\textbf{(D)}\ \text{only the fourth}\qquad
\textbf{(E)}\ \text{only the first}$
2010 AIME Problems, 11
Let $ \mathcal{R}$ be the region consisting of the set of points in the coordinate plane that satisfy both $ |8 \minus{} x| \plus{} y \le 10$ and $ 3y \minus{} x \ge 15$. When $ \mathcal{R}$ is revolved around the line whose equation is $ 3y \minus{} x \equal{} 15$, the volume of the resulting solid is $ \frac {m\pi}{n\sqrt {p}}$, where $ m$, $ n$, and $ p$ are positive integers, $ m$ and $ n$ are relatively prime, and $ p$ is not divisible by the square of any prime. Find $ m \plus{} n \plus{} p$.
2000 Harvard-MIT Mathematics Tournament, 8
Let $\vec{v_1},\vec{v_2},\vec{v_3},\vec{v_4}$ and $\vec{v_5}$ be vectors in three dimensions. Show that for some $i,j$ in $1,2,3,4,5$, $\vec{v_i}\cdot \vec{v_j}\ge 0$.
2014 Harvard-MIT Mathematics Tournament, 12
Find a nonzero monic polynomial $P(x)$ with integer coefficients and minimal degree such that $P(1-\sqrt[3]2+\sqrt[3]4)=0$. (A polynomial is called $\textit{monic}$ if its leading coefficient is $1$.)
1969 IMO Longlists, 39
$(HUN 6)$ Find the positions of three points $A,B,C$ on the boundary of a unit cube such that $min\{AB,AC,BC\}$ is the greatest possible.
2013 Romania National Olympiad, 1
The right prism $ABCA'B'C'$, with $AB = AC = BC = a$, has the property that there exists an unique point $M \in (BB')$ so that $AM \perp MC'$. Find the measure of the angle of the straight line $AM$ and the plane $(ACC')$ .
2004 IMC, 4
Suppose $n\geq 4$ and let $S$ be a finite set of points in the space ($\mathbb{R}^3$), no four of which lie in a plane. Assume that the points in $S$ can be colored with red and blue such that any sphere which intersects $S$ in at least 4 points has the property that exactly half of the points in the intersection of $S$ and the sphere are blue. Prove that all the points of $S$ lie on a sphere.
2001 AIME Problems, 15
Let $EFGH$, $EFDC$, and $EHBC$ be three adjacent square faces of a cube, for which $EC=8$, and let $A$ be the eighth vertex of the cube. Let $I$, $J$, and $K$, be the points on $\overline{EF}$, $\overline{EH}$, and $\overline{EC}$, respectively, so that $EI=EJ=EK=2$. A solid $S$ is obtained by drilling a tunnel through the cube. The sides of the tunnel are planes parallel to $\overline{AE}$, and containing the edges, $\overline{IJ}$, $\overline{JK}$, and $\overline{KI}$. The surface area of $S$, including the walls of the tunnel, is $m+n\sqrt{p}$, where $m$, $n$, and $p$ are positive integers and $p$ is not divisible by the square of any prime. Find $m+n+p$.
1979 IMO Longlists, 11
Prove that a pyramid $A_1A_2 \ldots A_{2k+1}S$ with equal lateral edges and equal space angles between adjacent lateral walls is regular.
1938 Moscow Mathematical Olympiad, 039
The following operation is performed over points $O_1, O_2, O_3$ and $A$ in space. The point $A$ is reflected with respect to $O_1$, the resultant point $A_1$ is reflected through $O_2$, and the resultant point $A_2$ through $O_3$. We get some point $A_3$ that we will also consecutively reflect through $O_1, O_2, O_3$. Prove that the point obtained last coincides with $A$..
2008 Bundeswettbewerb Mathematik, 3
Through a point in the interior of a sphere we put three pairwise perpendicular planes. Those planes dissect the surface of the sphere in eight curvilinear triangles. Alternately the triangles are coloured black and wide to make the sphere surface look like a checkerboard. Prove that exactly half of the sphere's surface is coloured black.
1967 IMO Shortlist, 3
Determine the volume of the body obtained by cutting the ball of radius $R$ by the trihedron with vertex in the center of that ball, it its dihedral angles are $\alpha, \beta, \gamma.$
1956 Polish MO Finals, 6
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]
2013 Online Math Open Problems, 44
Suppose tetrahedron $PABC$ has volume $420$ and satisfies $AB = 13$, $BC = 14$, and $CA = 15$. The minimum possible surface area of $PABC$ can be written as $m+n\sqrt{k}$, where $m,n,k$ are positive integers and $k$ is not divisible by the square of any prime. Compute $m+n+k$.
[i]Ray Li[/i]
1983 IMO Longlists, 67
The altitude from a vertex of a given tetrahedron intersects the opposite face in its orthocenter. Prove that all four altitudes of the tetrahedron are concurrent.
2021 Adygea Teachers' Geometry Olympiad, 4
Two identical balls of radius $\sqrt{15}$ and two identical balls of a smaller radius are located on a plane so that each ball touches the other three. Find the area of the surface $S$ of the ball with the smaller radius.
1980 Brazil National Olympiad, 4
Given $5$ points of a sphere radius $r$, show that two of the points are a distance $\le r \sqrt2$ apart.
1960 IMO Shortlist, 6
Consider a cone of revolution with an inscribed sphere tangent to the base of the cone. A cylinder is circumscribed about this sphere so that one of its bases lies in the base of the cone. let $V_1$ be the volume of the cone and $V_2$ be the volume of the cylinder.
a) Prove that $V_1 \neq V_2$;
b) Find the smallest number $k$ for which $V_1=kV_2$; for this case, construct the angle subtended by a diamter of the base of the cone at the vertex of the cone.
2009 Sharygin Geometry Olympiad, 22
Construct a quadrilateral which is inscribed and circumscribed, given the radii of the respective circles and the angle between the diagonals of quadrilateral.
2006 Tournament of Towns, 7
An ant craws along a closed route along the edges of a dodecahedron, never going backwards.
Each edge of the route is passed exactly twice. Prove that one of the edges is passed both times in the same direction. (Dodecahedron has $12$ faces in the shape of pentagon, $30$ edges and $20$ vertices; each vertex emitting 3 edges). (8)