Found problems: 2265
2008 ITest, 88
A six dimensional "cube" (a $6$-cube) has $64$ vertices at the points $(\pm 3,\pm 3,\pm 3,\pm 3,\pm 3,\pm 3).$ This $6$-cube has $192\text{ 1-D}$ edges and $240\text{ 2-D}$ edges. This $6$-cube gets cut into $6^6=46656$ smaller congruent "unit" $6$-cubes that are kept together in the tightly packaged form of the original $6$-cube so that the $46656$ smaller $6$-cubes share 2-D square faces with neighbors ($\textit{one}$ 2-D square face shared by $\textit{several}$ unit $6$-cube neighbors). How many 2-D squares are faces of one or more of the unit $6$-cubes?
2007 Princeton University Math Competition, 9
There are four spheres each of radius $1$ whose centers form a triangular pyramid where each side has length $2$. There is a 5th sphere which touches all four other spheres and has radius less than $1$. What is its radius?
2005 AMC 12/AHSME, 10
A wooden cube $ n$ units on a side is painted red on all six faces and then cut into $ n^3$ unit cubes. Exactly one-fourth of the total number of faces of the unit cubes are red. What is $ n$?
$ \textbf{(A)}\ 3\qquad
\textbf{(B)}\ 4\qquad
\textbf{(C)}\ 5\qquad
\textbf{(D)}\ 6\qquad
\textbf{(E)}\ 7$
2005 Federal Competition For Advanced Students, Part 2, 3
Let $Q$ be a point inside a cube. Prove that there are infinitely many lines $l$ so that $AQ=BQ$ where $A$ and $B$ are the two points of intersection of $l$ and the surface of the cube.
1980 IMO Shortlist, 15
Prove that the sum of the six angles subtended at an interior point of a tetrahedron by its six edges is greater than 540°.
1986 Traian Lălescu, 1.4
Let be two fixed points $ B,C. $ Find the locus of the spatial points $ A $ such that $ ABC $ is a nondegenerate triangle and the expression
$$ R^2 (A)\cdot\sin \left( 2\angle ABC\right)\cdot\sin \left( 2\angle BCA\right) $$
has the greatest value possible, where $ R(A) $ denotes the radius of the excirlce of $ ABC. $
1980 Bulgaria National Olympiad, Problem 6
Show that if all lateral edges of a pentagonal pyramid are of equal length and all the angles between neighboring lateral faces are equal, then the pyramid is regular.
1984 Polish MO Finals, 3
Let $W$ be a regular octahedron and $O$ be its center. In a plane $P$ containing $O$ circles $k_1(O,r_1)$ and $k_2(O,r_2)$ are chosen so that $k_1 \subset P\cap W \subset k_2$. Prove that $\frac{r_1}{r_2}\le \frac{\sqrt3}{2}$
1964 IMO Shortlist, 5
Supppose five points in a plane are situated so that no two of the straight lines joining them are parallel, perpendicular, or coincident. From each point perpendiculars are drawn to all the lines joining the other four points. Determine the maxium number of intersections that these perpendiculars can have.
1966 IMO Longlists, 56
In a tetrahedron, all three pairs of opposite (skew) edges are mutually perpendicular. Prove that the midpoints of the six edges of the tetrahedron lie on one sphere.
1982 Poland - Second Round, 6
Given a finite set $B$ of points in space, any two distances between the points of this set are different. Each point of the set $B$ is connected by a line segment to the closest point of the set $B$. This way we will get a set of sections, one of which (any chosen one) we paint red, all the remaining sections we paint green. Prove that there are two points of the set $B$ that cannot be connected by a line composed of green segments.
1982 IMO Shortlist, 18
Let $O$ be a point of three-dimensional space and let $l_1, l_2, l_3$ be mutually perpendicular straight lines passing through $O$. Let $S$ denote the sphere with center $O$ and radius $R$, and for every point $M$ of $S$, let $S_M$ denote the sphere with center $M$ and radius $R$. We denote by $P_1, P_2, P_3$ the intersection of $S_M$ with the straight lines $l_1, l_2, l_3$, respectively, where we put $P_i \neq O$ if $l_i$ meets $S_M$ at two distinct points and $P_i = O$ otherwise ($i = 1, 2, 3$). What is the set of centers of gravity of the (possibly degenerate) triangles $P_1P_2P_3$ as $M$ runs through the points of $S$?
2025 All-Russian Olympiad, 11.2
A right prism \(ABCA_1B_1C_1\) is given. It is known that triangles \(A_1BC\), \(AB_1C\), \(ABC_1\), and \(ABC\) are all acute. Prove that the orthocenters of these triangles, together with the centroid of triangle \(ABC\), lie on the same sphere.
2005 AMC 12/AHSME, 17
A unit cube is cut twice to form three triangular prisms, two of which are congruent, as shown in Figure 1. The cube is then cut in the same manner along the dashed lines shown in Figure 2. This creates nine pieces. What is the volume of the piece that contains vertex $ W$?
[asy]import three;
size(200);
defaultpen(linewidth(.8pt)+fontsize(10pt));
currentprojection=oblique;
path3 p1=(0,2,2)--(0,2,0)--(2,2,0)--(2,2,2)--(0,2,2)--(0,0,2)--(2,0,2)--(2,2,2);
path3 p2=(2,2,0)--(2,0,0)--(2,0,2);
path3 p3=(0,0,2)--(0,2,1)--(2,2,1)--(2,0,2);
path3 p4=(2,2,1)--(2,0,0);
pen finedashed=linetype("4 4");
draw(p1^^p2^^p3^^p4);
draw(shift((4,0,0))*p1);
draw(shift((4,0,0))*p2);
draw(shift((4,0,0))*p3);
draw(shift((4,0,0))*p4);
draw((4,0,2)--(5,2,2)--(6,0,2),finedashed);
draw((5,2,2)--(5,2,0)--(6,0,0),finedashed);
label("$W$",(3,0,2));
draw((2.7,.3,2)--(2.1,1.9,2),linewidth(.6pt));
draw((3.4,.3,2)--(5.9,1.9,2),linewidth(.6pt));
label("Figure 1",(1,-0.5,2));
label("Figure 2",(5,-0.5,2));[/asy]$ \textbf{(A)}\ \frac {1}{12}\qquad \textbf{(B)}\ \frac {1}{9}\qquad \textbf{(C)}\ \frac {1}{8}\qquad \textbf{(D)}\ \frac {1}{6}\qquad \textbf{(E)}\ \frac {1}{4}$
2014 Bundeswettbewerb Mathematik, 2
The $100$ vertices of a prism, whose base is a $50$-gon, are labeled with numbers $1, 2, 3, \ldots, 100$ in any order. Prove that there are two vertices, which are connected by an edge of the prism, with labels differing by not more than $48$.
Note: In all the triangles the three vertices do not lie on a straight line.
1974 IMO Longlists, 44
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.
III Soros Olympiad 1996 - 97 (Russia), 11.10
In a dihedral angle of measure $c$ two non-intersecting spheres are inscribed, the centers of which are located on a straight line perpendicular to the edge of the dihedral angle. The points of contact of these spheres with the edges of the corner are at distances $a$ and $b$ from the edge. Let us consider an arbitrary plane tangent to these spheres and intersecting the segment connecting their centers. Let us denote by $\phi$ the measure of the angle formed at the intersection of this plane with the faces of a given dihedral angle. Find the greatest value $\phi$.
2002 May Olympiad, 1
Using white cubes of side $1$, a prism (without holes) was assembled. The faces of the prism were painted black. It is known that the cubes left with exactly $4$ white faces are $20$ in total. Determine what the dimensions of the prism can be. Give all the possibilities.
1971 IMO Longlists, 49
Let $P_1$ be a convex polyhedron with vertices $A_1,A_2,\ldots,A_9$. Let $P_i$ be the polyhedron obtained from $P_1$ by a translation that moves $A_1$ to $A_i$. Prove that at least two of the polyhedra $P_1,P_2,\ldots,P_9$ have an interior point in common.
1994 French Mathematical Olympiad, Problem 2
Let be given a semi-sphere $\Sigma$ whose base-circle lies on plane $p$. A variable plane $Q$, parallel to a fixed plane non-perpendicular to $P$, cuts $\Sigma$ at a circle $C$. We denote by $C'$ the orthogonal projection of $C$ onto $P$. Find the position of $Q$ for which the cylinder with bases $C$ and $C'$ has the maximum volume.
2022 JHMT HS, 9
Let $B$ and $D$ be two points chosen independently and uniformly at random from the unit sphere in 3D space centered at a point $A$ (this unit sphere is the set of all points in $\mathbb{R}^3$ a distance of $1$ away from $A$). Compute the expected value of $\sin^2\angle DAB$.
2013 Miklós Schweitzer, 6
Let ${\mathcal A}$ be a ${C^{\ast}}$ algebra with a unit element and let ${\mathcal A_+}$ be the cone of the positive elements of ${\mathcal A}$ (this is the set of such self adjoint elements in ${\mathcal A}$ whose spectrum is in ${[0,\infty)}$. Consider the operation
\[ \displaystyle x \circ y =\sqrt{x}y\sqrt{x},\ x,y \in \mathcal A_+\]
Prove that if for all ${x,y \in \mathcal A_+}$ we have
\[ \displaystyle (x\circ y)\circ y = x \circ (y \circ y), \]
then ${\mathcal A}$ is commutative.
[i]Proposed by Lajos Molnár[/i]
1971 Poland - Second Round, 3
There are 6 lines in space, of which no 3 are parallel, no 3 pass through the same point, and no 3 are contained in the same plane. Prove that among these 6 lines there are 3 mutually oblique lines.
1985 Polish MO Finals, 6
There is a convex polyhedron with $k$ faces.
Show that if more than $k/2$ of the faces are such that no two have a common edge,
then the polyhedron cannot have an inscribed sphere.
1956 Putnam, B3
A sphere is inscribed in a tetrahedron and each point of contact of the sphere with the four faces is joined to the vertices of the face containing the point. Show that the four sets of three angles so formed are identical.