Found problems: 2265
In the figure, polygons $ A$, $ E$, and $ F$ are isosceles right triangles; $ B$, $ C$, and $ D$ are squares with sides of length $ 1$; and $ G$ is an equilateral triangle. The figure can be folded along its edges to form a polyhedron having the polygons as faces. The volume of this polyhedron is
$ \textbf{(A)}\ 1/2\qquad \textbf{(B)}\ 2/3\qquad \textbf{(C)}\ 3/4\qquad \textbf{(D)}\ 5/6\qquad \textbf{(E)}\ 4/3$
[asy]
size(180);
defaultpen(linewidth(.7pt)+fontsize(10pt));
draw((-1,1)--(2,1));
draw((-1,0)--(1,0));
draw((-1,1)--(-1,0));
draw((0,-1)--(0,3));
draw((1,2)--(1,0));
draw((-1,1)--(1,1));
draw((0,2)--(1,2));
draw((0,3)--(1,2));
draw((0,-1)--(2,1));
draw((0,-1)--((0,-1) + sqrt(2)*dir(-15)));
draw(((0,-1) + sqrt(2)*dir(-15))--(1,0));
label("$\textbf{A}$",foot((0,2),(0,3),(1,2)),SW);
label("$\textbf{B}$",midpoint((0,1)--(1,2)));
label("$\textbf{C}$",midpoint((-1,0)--(0,1)));
label("$\textbf{D}$",midpoint((0,0)--(1,1)));
label("$\textbf{E}$",midpoint((1,0)--(2,1)),NW);
label("$\textbf{F}$",midpoint((0,-1)--(1,0)),NW);
label("$\textbf{G}$",midpoint((0,-1)--(1,0)),2SE);[/asy]
Each face of a tetrahedron can be placed in a circle of radius $ 1$. Show that the tetrahedron can be placed in a sphere of radius $ \frac{3}{2\sqrt2}$.
Faces of a convex polyhedron are six squares and 8 equilateral triangles and each edge is a common side for one triangle and one square. All dihedral angles obtained from the triangle and square with a common edge, are equal. Prove that it is possible to circumscribe a sphere around the polyhedron, and compute the ratio of the squares of volumes of that polyhedron and of the ball whose boundary is the circumscribed sphere.
Let $OABC$ be a tetrahedon with $\angle BOC=\alpha,\angle COA =\beta$ and $\angle AOB =\gamma$. The angle between the faces $OAB$ and $OAC$ is $\sigma$ and the angle between the faces $OAB$ and $OBC$ is $\rho$.
Show that $\gamma > \beta \cos\sigma + \alpha \cos\rho$.
In tetrahedron $ABCD$ with circumradius $2$, $AB = 2$, $CD = \sqrt{7}$, and $\angle ABC = \angle BAD = \frac{\pi}{2}$. Find all possible angles between the planes containing $ABC$ and $ABD$.
[b]U[/b]ndecillion years ago in a galaxy far, far away, there were four space stations in the three-dimensional space, each pair spaced 1 light year away from each other. Admiral Ackbar wanted to establish a base somewhere in space such that the sum of squares of the distances from the base to each of the stations does not exceed 15 square light years. (The sizes of the space stations and the base are negligible.) Determine the volume, in cubic light years, of the set of all possible locations for the Admiral’s base.
The points $P$ and $Q$ are the midpoints of the edges $AE$ and $CG$ on the cube $ABCD.EFGH$ respectively. If the length of the cube edges is $1$ unit, determine the area of the quadrilateral $DPFQ$ .
Prove that in space there is a sphere containing exactly $1994$ points with integer coordinates.
A sphere with center at $O$ is inscribed in a trihedral angle with vertex $S$. Prove that the plane passing through the three tangent points is perpendicular to $OS$.
Find all $3$-digit numbers which are the sums of the cubes of their digits.
If five geometric means are inserted between 8 and 5832, the fifth term in the geometric series:
$\textbf{(A)}\ 648 \qquad
\textbf{(B)}\ 832 \qquad
\textbf{(C)}\ 1168 \qquad
\textbf{(D)}\ 1944 \qquad
\textbf{(E)}\ \text{None of these}$
A $ 9\times9\times9$ cube is composed of twenty-seven $ 3\times3\times3$ cubes. The big cube is 'tunneled' as follows: First, the six $ 3\times3\times3$ cubes which make up the center of each face as well as the center of $ 3\times3\times3$ cube are removed. Second, each of the twenty remaining $ 3\times3\times3$ cubes is diminished in the same way. That is, the central facial unit cubes as well as each center cube are removed.
[asy]
import three;
size(4.5cm);
triple eye = (6, 9, 5);
currentprojection = perspective(eye);
real eps = 0.001;
for(int i = 0; i < 3; ++i){
for(int j = 0; j < 3; ++j){
for(int k = 0; k < 3; ++k){
if(i == 1 && j == 1) continue;
if(j == 1 && k == 1) continue;
if(k == 1 && i == 1) continue;
draw(shift(i, j, k) * scale(1 - eps, 1 - eps, 1 - eps) * unitcube, gray(0.9), nolight);
draw(shift(i, j, k) * (X--(X + Y)--Y--(Y+Z)--Z--(Z + X)--cycle));
draw(shift(i, j, k) * (X + Y + Z--X + Y));
draw(shift(i, j, k) * (X + Y + Z--Y + Z));
draw(shift(i, j, k) * (X + Y + Z--Z + X));
}
}
}
[/asy]
The surface area of the final figure is
$ \textbf{(A)}\ 384\qquad
\textbf{(B)}\ 729\qquad
\textbf{(C)}\ 864\qquad
\textbf{(D)}\ 1024\qquad
\textbf{(E)}\ 1056$
Assume that it is possible to color more than half of the surfaces of a given polyhedron so that no two colored surfaces have a common edge.
(a) Describe one polyhedron with the above property.
(b) Prove that one cannot inscribe a sphere touching all the surfaces of a polyhedron with the above property.
A tetrahedron $OA_1B_1C_1$ is given. Let $A_2,A_3 \in OA_1, A_2,A_3 \in OA_1, A_2,A_3 \in OA_1$ be points such that the planes $A_1B_1C_1,A_2B_2C_2$ and $A_3B_3C_3$ are parallel and $OA_1 > OA_2 > OA_3 > 0$. Let $V_i$ be the volume of the tetrahedron $OA_iB_iC_i$ ($i = 1,2,3$) and $V$ be the volume of $OA_1B_2C_3$. Prove that $V_1 +V_2 +V_3 \ge 3V$.
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]
Let $S$ be a fixed point on a sphere $\Sigma$ with center $\Omega$. Consider all tetrahedra $SABC$ inscribed in $\Sigma$ such that $SA,SB,SC$ are pairwise orthogonal.
(a) Prove that all the planes $ABC$ pass through a single point.
(b) In one such tetrahedron, $H$ and $O$ are the orthogonal projections of $S$ and $\Omega$ onto the plane $ABC$, respectively. Let $R$ denote the circumradius of $\triangle ABC$. Prove that $R^2=OH^2+2SH^2$.
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.
The height and radius of the base of the cylinder are equal to $1$. What is the smallest number of balls of radius $1$ that can cover the entire cylinder?
A regular icosahedron is a regular solid of $20$ faces, each in the form of an equilateral triangle, with $12$ vertices, so that each vertex is in $5$ edges.
Twelve indistinguishable candies are glued to the vertices of a regular icosahedron (one at each vertex), and four of these twelve candies are special. André and Lucas want to together create a strategy for the following game:
• First, André is told which are the four special sweets and he must remove exactly four sweets that are not special from the icosahedron and leave the solid on a table, leaving afterwards without communicating with Lucas.
• Later, Sponchi, who wants to prevent Lucas from discovering the special sweets, can pick up the icosahedron from the table and rotate it however he wants.
• After Sponchi makes his move, he leaves the room, Lucas enters and he must determine the four special candies out of the eight that remain in the icosahedron.
Determine if there is a strategy for which Lucas can always properly discover the four special sweets.
Seven tetrahedra are placed on the table. For any three of them there exists a horizontal plane cutting them in triangles of equal areas. Show that there exists a plane cutting all seven tetrahedra in triangles of equal areas.
A pyramid $MABCD$ with the top-vertex $M$ is circumscribed about a sphere with center $O$ so that $O$ lies on the altitude of the pyramid. Each of the planes $ACM,BDM,ABO$ divides the lateral surface of the pyramid into two parts of equal areas. The areas of the sections of the planes $ACM$ and $ABO$ inside the pyramid are in ratio $(\sqrt2+2):4$. Determine the angle $\delta$ between the planes $ACM$ and $ABO$, and the dihedral angle of the pyramid at the edge $AB$.
How many positive integer cubes divide $ 3!\cdot 5!\cdot 7!$?
$ \textbf{(A)}\ 2\qquad
\textbf{(B)}\ 3\qquad
\textbf{(C)}\ 4\qquad
\textbf{(D)}\ 5\qquad
\textbf{(E)}\ 6$
The sphere $ \omega $ passes through the vertex $S$ of the pyramid $SABC$ and intersects with the edges $SA,SB,SC$ at $A_1,B_1,C_1$ other than $S$. The sphere $ \Omega $ is the circumsphere of the pyramid $SABC$ and intersects with $ \omega $ circumferential, lies on a plane which parallel to the plane $(ABC)$.
Points $A_2,B_2,C_2$ are symmetry points of the points $A_1,B_1,C_1$ respect to midpoints of the edges $SA,SB,SC$ respectively. Prove that the points $A$, $B$, $C$, $A_2$, $B_2$, and $C_2$ lie on a sphere.
Ants, named Anna and Bob, are located at vertices \(A\) and \(B\) respectively of a cube \(ABCD A_1 B_1 C_1 D_1\), with a sugar cube placed at vertex \(C_1\). It is known that Bob can move at a speed of $20$ meters per minute. Determine the minimum speed in integer meters per minute that Anna must be able to travel in order to reach the sugar cube at \(C_1\) before Bob.
[i]Proposed by Tamar Turashvili, Georgia [/i]
The chords $AB$ and $CD$ of a sphere intersect at $X. A, C$ and $X$ are equidistant from a point $Y$ on the sphere. Show that $BD$ and $XY$ are perpendicular.