Found problems: 2265
In a tetrahedron $ABCD$ the edges $AB$ and $CD$ are perpendicular and $\angle ACB =\angle ADB$. Prove that the plane through $AB$ and the midpoint of the edge $CD$, is perpendicular to $CD$.
Six spheres of radius $1$ are positioned so that their centers are at the vertices of a regular hexagon of side length $2$. The six spheres are internally tangent to a larger sphere whose center is the center of the hexagon. An eighth sphere is externally tangent to the six smaller spheres and internally tangent to the larger sphere. What is the radius of this eighth sphere?
$ \textbf{(A)} \ \sqrt{2} \qquad \textbf{(B)} \ \frac{3}{2} \qquad \textbf{(C)} \ \frac{5}{3} \qquad \textbf{(D)} \ \sqrt{3} \qquad \textbf{(E)} \ 2$
The faces of a cube are painted in six different colors: red (R), white (W), green (G), brown (B), aqua (A), and purple (P). Three views of the cube are shown below. What is the color of the face opposite the aqua face?
$\text{\textbf{(A) }red}\qquad \text{\textbf{(B) } white}\qquad\text{\textbf{(C) }green}\qquad\text{\textbf{(D) }brown }\qquad\text{\textbf{(E)} purple}$
[asy]
unitsize(2 cm);
pair x, y, z, trans;
int i;
x = dir(-5);
y = (0.6,0.5);
z = (0,1);
trans = (2,0);
for (i = 0; i <= 2; ++i) {
draw(shift(i*trans)*((0,0)--x--(x + y)--(x + y + z)--(y + z)--z--cycle));
draw(shift(i*trans)*((x + z)--x));
draw(shift(i*trans)*((x + z)--(x + y + z)));
draw(shift(i*trans)*((x + z)--z));
}
label(rotate(-3)*"$R$", (x + z)/2);
label(rotate(-5)*slant(0.5)*"$B$", ((x + z) + (y + z))/2);
label(rotate(35)*slant(0.5)*"$G$", ((x + z) + (x + y))/2);
label(rotate(-3)*"$W$", (x + z)/2 + trans);
label(rotate(50)*slant(-1)*"$B$", ((x + z) + (y + z))/2 + trans);
label(rotate(35)*slant(0.5)*"$R$", ((x + z) + (x + y))/2 + trans);
label(rotate(-3)*"$P$", (x + z)/2 + 2*trans);
label(rotate(-5)*slant(0.5)*"$R$", ((x + z) + (y + z))/2 + 2*trans);
label(rotate(-85)*slant(-1)*"$G$", ((x + z) + (x + y))/2 + 2*trans);
[/asy]
Given four points $A,B,C,D$ in space, find a plane, $S$, equidistant from all four points and having $A$ and $C$ on one side, $B$ and $D$ on the other.
Prove that the group of orientation-preserving symmetries of the cube is isomorphic to $S_4$ (the group of permutations of $\{1,2,3,4\}$).(20 points)
A $4\times4\times4$ cube is composed of $64$ unit cubes. The faces of $16$ unit cubes are to be coloured red. A colouring is called interesting if there is exactly $1$ red unit cube in every $1\times1\times 4$ rectangular box composed of $4$ unit cubes. Determine the number of interesting colourings.
(a) Is it possible to place five wooden cubes in space so that each of them has a part of its face touching each of the others?
(b) Answer the same question, but with $6$ cubes.
Show that each integer $n$ can be written as the sum of five perfect cubes (not necessarily positive).
Calculate the volume $ V $ of tetrahedron $ ABCD $ given the length $ d $ of edge $ AB $ and the area $ S $ of the projection of the tetrahedron on the plane perpendicular to the line $ AB $.
In a tetrahedron $ABCD$, the four altitudes are concurrent at $H$. The line $DH$ intersects the plane $ABC$ at $P$ and the circumsphere of $ABCD$ at $Q\neq D$. Prove that $PQ=2HP$.
Find an ordered pair $(a,b)$ of real numbers for which $x^2+ax+b$ has a non-real root whose cube is $343$.
Does there exist a regular triangular prism that can be covered (without overlapping) by different equilateral triangles? (One is allowed to bend the triangles around the edges of the prism.)
In tetrahedron $ABCD$, vertex $D$ is connected with $D_0$, the centrod if $\triangle ABC$. Line parallel to $DD_0$ are drawn through $A,B$ and $C$. These lines intersect the planes $BCD, CAD$ and $ABD$ in points $A_2, B_1,$ and $C_1$, respectively. Prove that the volume of $ABCD$ is one third the volume of $A_1B_1C_1D_0$. Is the result if point $D_o$ is selected anywhere within $\triangle ABC$?
Let $ABCD$ be a regular tetrahedron. To an arbitrary point $M$ on one edge, say $CD$, corresponds the point $P = P(M)$ which is the intersection of two lines $AH$ and $BK$, drawn from $A$ orthogonally to $BM$ and from $B$ orthogonally to $AM$. What is the locus of $P$ when $M$ varies ?
A sphere is inscribed into a prism $ABCA'B'C'$ and touches its lateral faces $BCC'B', CAA'C', ABB'A' $ at points $A_o, B_o, C_o$ respectively. It is known that $\angle A_oBB' = \angle B_oCC' =\angle C_oAA'$.
a) Find all possible values of these angles.
b) Prove that segments $AA_o, BB_o, CC_o$ concur.
c) Prove that the projections of the incenter to $A'B', B'C', C'A'$ are the vertices of a regular triangle.
A regular pentagon $A_1A_2A_3A_4A_5$ with side length $s$ is given. At each point $A_i$, a sphere $K_i$ of radius $\frac{s}{2}$ is constructed. There are two spheres $K_1$ and $K_2$ each of radius $\frac{s}{2}$ touching all the five spheres $K_i.$ Decide whether $K_1$ and $K_2$ intersect each other, touch each other, or have no common points.
We consider a spherical bowl without any great circle. The distance between points $A$ and $B$ on such a bowl is defined as the length of the arc of the great circle of the sphere with ends at points $A$ and $B$, which is contained in the bowl. Prove that there is no isometry mapping this bowl to a subset of the plane.
Attention. A spherical bowl is each of the two parts into which the surface of the sphere is divided by a plane intersecting the sphere.
Find the positive integers $n$ that are not divisible by $3$ if the number $2^{n^2-10}+2133$ is a perfect cube.
[hide="Note"]
[color=#BF0000]The wording of this problem is perhaps not the best English. As far as I am aware, just solve the diophantine equation $x^3=2^{n^2-10}+2133$ where $x,n \in \mathbb{N}$ and $3\nmid n$.[/color][/hide]
Given a tetrahedron $ T$. Valentin wants to find two its edges $ a,b$ with no common vertices so that $ T$ is covered by balls with diameters $ a,b$. Can he always find such a pair?
[i]A. Zaslavsky[/i]
A $10 \times 10 \times 10$ cube is made up up from $500$ white unit cubes and $500$ black unit cubes, arranged in such a way that every two unit cubes that shares a face are in different colors. A line is a $1 \times 1 \times 10$ portion of the cube that is parallel to one of cube’s edges. From the initial cube have been removed $100$ unit cubes such that $300$ lines of the cube has exactly one missing cube.
Determine if it is possible that the number of removed black unit cubes is divisible by $4$.
Which regular polygon can be obtained (and how) by cutting a cube with a plane ?
Let $ABCDA'B'C'D'$ be a rectangular prism with $|AB|=2|BC|$. $E$ is a point on the edge $[BB']$ satisfying $|EB'|=6|EB|$. Let $F$ and $F'$ be the feet of the perpendiculars from $E$ at $\triangle AEC$ and $\triangle A'EC'$, respectively. If $m(\widehat{FEF'})=60^{\circ}$, then $|BC|/|BE| = ? $
$ \textbf{(A)}\ \sqrt\frac53 \qquad \textbf{(B)}\ \sqrt\frac{15}2 \qquad \textbf{(C)}\ \frac32\sqrt{15} \qquad \textbf{(D)}\ 5\sqrt\frac53 \qquad \textbf{(E)}\ \text{None}$
A set $ S$ of points from the space will be called [b]completely symmetric[/b] if it has at least three elements and fulfills the condition that for every two distinct points $ A$ and $ B$ from $ S$, the perpendicular bisector plane of the segment $ AB$ is a plane of symmetry for $ S$. Prove that if a completely symmetric set is finite, then it consists of the vertices of either a regular polygon, or a regular tetrahedron or a regular octahedron.
Let $A$, $B$, $C$, and $D$ be the vertices of a regular tetrahedron, each of whose edges measures 1 meter. A bug, starting from vertex $A$, observes the following rule: at each vertex it chooses one of the three edges meeting at that vertex, each edge being equally likely to be chosen, and crawls along that edge to the vertex at its opposite end. Let $p = n/729$ be the probability that the bug is at vertex $A$ when it has crawled exactly 7 meters. Find the value of $n$.
Consider a cube and two of its vertices $A$ and $B$, which are the endpoints of a face diagonal. A [i]path [/i] is a sequence of cube angles, each step of one angle along a cube edge is walked to one of the three adjacent angles. Let $a$ be the number of paths of length $2012$ that starts at point $A$ and ends at $A$ and let b be the number of ways of length $2012$ that starts in $A$ and ends in $B$. Decide which of the two numbers $a$ and $b$ is the larger.