Found problems: 2265
A plastic snap-together cube has a protruding snap on one side and receptacle holes on the other five sides as shown. What is the smallest number of these cubes that can be snapped together so that only receptacle holes are showing?
[asy]
draw((0,0)--(4,0)--(4,4)--(0,4)--cycle);
draw(circle((2,2),1));
draw((4,0)--(6,1)--(6,5)--(4,4));
draw((6,5)--(2,5)--(0,4));
draw(ellipse((5,2.5),0.5,1));
fill(ellipse((3,4.5),1,0.25),black);
fill((2,4.5)--(2,5.25)--(4,5.25)--(4,4.5)--cycle,black);
fill(ellipse((3,5.25),1,0.25),black);
[/asy]
$\text{(A)}\ 3 \qquad \text{(B)}\ 4 \qquad \text{(C)}\ 5 \qquad \text{(D)}\ 6 \qquad \text{(E)}\ 8$
A triangular pyramid $ O.ABC$ with base $ ABC$ has the property that the lengths of the altitudes from $ A$, $ B$ and $ C$ are not less than $ \frac{OB \plus{}OC}{2}$, $ \frac{OC \plus{} OA}{2}$ and $ \frac{OA \plus{} OB}{2}$, respectively. Given that the area of $ ABC$ is $ S$, calculate the volume of the pyramid.
Prove that the six sides of any tetrahedron can be the sides of a convex hexagon.
Let be given an arbitrary tetrahedron $ABCD$ with volume $V$. Consider all lines which pass through the barycenter $S$ of the tetrahedron and intersect the edges $AD,BD,CD$ at points $A',B',C$ respectively. It is known that among the obtained tetrahedra there exists one with the minimal volume. Express this minimal volume in terms of $V$
Triangle $ ABC$, with sides of length $ 5$, $ 6$, and $ 7$, has one vertex on the positive $ x$-axis, one on the positive $ y$-axis, and one on the positive $ z$-axis. Let $ O$ be the origin. What is the volume of tetrahedron $ OABC$?
$ \textbf{(A)}\ \sqrt{85} \qquad
\textbf{(B)}\ \sqrt{90} \qquad
\textbf{(C)}\ \sqrt{95} \qquad
\textbf{(D)}\ 10 \qquad
\textbf{(E)}\ \sqrt{105}$
The points $A, B$ and $C$ lie on the surface of a sphere with center $O$ and radius 20. It is given that $AB=13, BC=14, CA=15,$ and that the distance from $O$ to triangle $ABC$ is $\frac{m\sqrt{n}}k,$ where $m, n,$ and $k$ are positive integers, $m$ and $k$ are relatively prime, and $n$ is not divisible by the square of any prime. Find $m+n+k.$
For what digit $A$ is the numeral $1AA$ a perfect square in base-$5$ and a perfect cube in base-$6$?
$\text{(A) }0\qquad\text{(B) }1\qquad\text{(C) }2\qquad\text{(D) }3\qquad\text{(E) }4$
Fourteen white cubes are put together to form the fi gure on the right. The complete surface of the figure, including the bottom, is painted red. The figure is then separated into individual cubes. How many of the individual cubes have exactly four red faces?
[asy]
import three;
defaultpen(linewidth(0.8));
real r=0.5;
currentprojection=orthographic(3/4,8/15,7/15);
draw(unitcube, white, thick(), nolight);
draw(shift(1,0,0)*unitcube, white, thick(), nolight);
draw(shift(2,0,0)*unitcube, white, thick(), nolight);
draw(shift(0,0,1)*unitcube, white, thick(), nolight);
draw(shift(2,0,1)*unitcube, white, thick(), nolight);
draw(shift(0,1,0)*unitcube, white, thick(), nolight);
draw(shift(2,1,0)*unitcube, white, thick(), nolight);
draw(shift(0,2,0)*unitcube, white, thick(), nolight);
draw(shift(2,2,0)*unitcube, white, thick(), nolight);
draw(shift(0,3,0)*unitcube, white, thick(), nolight);
draw(shift(0,3,1)*unitcube, white, thick(), nolight);
draw(shift(1,3,0)*unitcube, white, thick(), nolight);
draw(shift(2,3,0)*unitcube, white, thick(), nolight);
draw(shift(2,3,1)*unitcube, white, thick(), nolight);[/asy]
$ \textbf{(A)}\ 4\qquad\textbf{(B)}\ 6\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 12$
Let $O$ and $O'$ designate two dìagonally opposite vertices of a cube. Bisect those edges of the cube that contain neither of the points $O$ and $O'$. Prove that these midpoints of edges lie in a plane and form the vertices of a regular hexagon
The four bottom corners of a cube are colored red, green, blue, and purple. How many ways are there to color the top four corners of the cube so that every face has four different colored corners? Prove that your answer is correct.
The real root of the equation $8x^3 - 3x^2 - 3x - 1 = 0$ can be written in the form $\frac{\sqrt[3]a + \sqrt[3]b + 1}{c}$, where $a$, $b$, and $c$ are positive integers. Find $a+b+c$.
For a positive real number $t$, in the coordiante space, consider 4 points $O(0,\ 0,\ 0),\ A(t,\ 0,\ 0),\ B(0,\ 1,\ 0),\ C(0,\ 0,\ 1)$.
Let $r$ be the radius of the sphere $P$ which is inscribed to all faces of the tetrahedron $OABC$.
When $t$ moves, find the maximum value of $\frac{\text{vol[P]}}{\text{vol[OABC]}}.$
Prove that every cube's cross-section, containing its centre, has the area not less then its face's area.
In a metallic disk, a circular sector is removed, so that with the remaining can form a conical glass of maximum volume. Calculate, in radians, the angle of the sector that is removed.
[hide=original wording]En un disco metalico se quita un sector circular, de modo que con la parte restante se pueda formar un vaso c´onico de volumen maximo. Calcular, en radianes, el angulo del sector que se quita.[/hide]
Title is self explanatory. Pick two points on the unit sphere. What is the expected distance between them?
Given a tetrahedron $VABC$ with edges $VA$, $VB$ and $VC$ perpendicular any two of them. The sum of the lengths of the tetrahedron's edges is $3p$. Find the maximal volume of $VABC$.
For each of $k=1,2,3,4,5$ find necessary and sufficient conditions on $a>0$ such that there exists a tetrahedron with $k$ edges length $a$ and the remainder length $1$.
Consider the regular quadrilateral prism $ABCDA'B'C 'D'$, in which $AB = a,AA' = \frac{a \sqrt {2}}{2}$, and $M$ is the midpoint of $B' C'$. Let $F$ be the foot of the perpendicular from $B$ on line $MC$, Let determine the measure of the angle between the planes $(BDF)$ and $(HBS)$.
Prove that the sum of the six angles subtended at an interior point of a tetrahedron by its six edges is greater than 540°.
A cube of integral dimensions is given in space so that all four vertices of one of the faces are lattice points. Prove that the other four vertices are also lattice points.
Given three points in space forming an acute-angled triangle, show that we can find two further points such that no three of the five points are collinear and the line through any two is normal to the plane through the other three.
The orthogonal projections of the vertices $A, B, C$ of the tetrahedron $ABCD$ on the opposite faces are denoted by $A', B', C'$ respectively. Suppose that point $A'$ is the circumcenter of the triangle $BCD$, point $B'$ is the incenter of the triangle $ACD$ and $C'$ is the centroid of the triangle $ABD$. Prove that tetrahedron $ABCD$ is regular.
A convex polyhedron $ Q$ has vertices $ V_1,V_2,\ldots,V_n$, and $ 100$ edges. The polyhedron is cut by planes $ P_1,P_2,\ldots,P_n$ in such a way that plane $ P_k$ cuts only those edges that meet at vertex $ V_k$. In addition, no two planes intersect inside or on $ Q$. The cuts produce $ n$ pyramids and a new polyhedron $ R$. How many edges does $ R$ have?
$ \textbf{(A)}\ 200\qquad
\textbf{(B)}\ 2n\qquad
\textbf{(C)}\ 300\qquad
\textbf{(D)}\ 400\qquad
\textbf{(E)}\ 4n$
Given three congruent rectangles in the space. Their centers coincide, but the planes they lie in are mutually perpendicular. For any two of the three rectangles, the line of intersection of the planes of these two rectangles contains one midparallel of one rectangle and one midparallel of the other rectangle, and these two midparallels have different lengths. Consider the convex polyhedron whose vertices are the vertices of the rectangles.
[b]a.)[/b] What is the volume of this polyhedron ?
[b]b.)[/b] Can this polyhedron turn out to be a regular polyhedron ? If yes, what is the condition for this polyhedron to be regular ?
Prove that every polyhedron has at least two faces with the same number of sides.