Found problems: 2265
A $ 3 \times 3 \times 3$ cube is formed by gluing together 27 standard cubical dice. (On a standard die, the sum of the numbers on any pair of opposite faces is 7.) The smallest possible sum of all the numbers showing on the surface of the $ 3 \times 3 \times 3$ cube is
$ \text{(A)}\ 60 \qquad
\text{(B)}\ 72 \qquad
\text{(C)}\ 84 \qquad
\text{(D)}\ 90 \qquad
\text{(E)}\ 96$
Consider skew lines $PP'$, $QQ'$ and points $X$, $Y$ lying on them, respectively. Initially, we have $X=P$, $Y=Q$. Both points $X$, $Y$ start moving simultaneously along the rays $PP'$, $QQ'$ with the speeds $c_1$, $c_2$, respectively. Show that midpoint $Z$ of segment $XY$ always lies on a fixed ray $RR'$, where $R$ is midpoint of $PQ$.
There is three-dimensional space. For every integer $n$ we build planes $ x \pm y\pm z = n$. All space is divided on octahedrons and tetrahedrons.
Point $(x_0,y_0,z_0)$ has rational coordinates but not lies on any plane. Prove, that there is such natural $k$ , that point $(kx_0,ky_0,kz_0)$ lies strictly inside the octahedron of partition.
Four faces of tetrahedron $ABCD$ are congruent triangles whose angles form an arithmetic progression. If the lengths of the sides of the triangles are $a < b < c$, determine the radius of the sphere circumscribed about the tetrahedron as a function on $a, b$, and $c$. What is the ratio $c/a$ if $R = a \ ?$
In the adjoining figure, $PQ$ is a diagonal of the cube. If $PQ$ has length $a$, then the surface area of the cube is
$\text{(A)}\ 2a^2 \qquad \text{(B)}\ 2\sqrt{2}a^2 \qquad \text{(C)}\ 2\sqrt{3}a^2 \qquad \text{(D)}\ 3\sqrt{3}a^2 \qquad \text{(E)}\ 6a^2$
Consider the sums of the form $\sum_{k=1}^{n} \epsilon_k k^3,$ where $\epsilon_k \in \{-1, 1\}.$ Is any of these sums equal to $0$ if
[b](a)[/b] $n=2000;$
[b](b)[/b] $n=2001 \ ?$
In a right circular cone with the radius of the base $R$ and the height $h$ are $n$ spheres of the same radius $r$ ($n \ge 3$). Each ball touches the base of the cone, its side surface and other two balls. Determine $r$.
A regular triangular prism has the altitude $h,$ and the two bases of the prism are equilateral triangles with side length $a.$ Dream-holes are made in the centers of both bases, and the three lateral faces are mirrors. Assume that a ray of light, entering the prism through the dream-hole in the upper base, then being reflected once by any of the three mirrors, quits the prism through the dream-hole in the lower base. Find the angle between the upper base and the light ray at the moment when the light ray entered the prism, and the length of the way of the light ray in the interior of the prism.
Let $a_{ij}$ $i=1,2,3$; $j=1,2,3$ be real numbers such that $a_{ij}$ is positive for $i=j$ and negative for $i\neq j$.
Prove the existence of positive real numbers $c_{1}$, $c_{2}$, $c_{3}$ such that the numbers \[a_{11}c_{1}+a_{12}c_{2}+a_{13}c_{3},\qquad a_{21}c_{1}+a_{22}c_{2}+a_{23}c_{3},\qquad a_{31}c_{1}+a_{32}c_{2}+a_{33}c_{3}\] are either all negative, all positive, or all zero.
[i]Proposed by Kiran Kedlaya, USA[/i]
Let $v$ be a vertex of a cube $C$ with edges of length $4$. Let $S$ be the largest sphere that can be inscribed in $C$. Let $R$ be the region consisting of all points $p$ between $S$ and $C$ such that $p$ is closer to $v$ than to any other vertex of the cube. Find the volume of $R$.
Find all real numbers with the following property: the difference of its cube and
its square is equal to the square of the difference of its square and the number itself.
An ant is at vertex of a cube. Every $10$ minutes it moves to an adjacent vertex along an edge. If $N$ is the number of one hour journeys that end at the starting vertex, find the sum of the squares of the digits of $N$.
A convex polyhedron has $2n$ faces ($n\ge 3$), and all faces are triangles. What is the largest number of vertices at which converges exactly $3$ edges at a such a polyhedron ?
(a) A point $A$ is marked inside a sphere. Three perpendicular lines drawn through $A$ intersect the sphere at six points. Prove that the centre of gravity of these six points does not depend on the choice of such three lines.
(b) An icosahedron with the centre $A$ is placed inside a sphere (its centre does not necessarily coincide with the centre of the sphere). The rays going from $A$ to the vertices of the icosahedron mark $12$ points on the sphere. Then the icosahedron is rotated about its centre. New rays mark new $12$ points on the sphere. Let $O$ and $N$ be the centres of mass of old and new points respectively. Prove that $O = N$.
What is the greatest number of balls of radius $1/2$ that can be placed within a rectangular box of size $10 \times 10 \times 1 \ ?$
(V.Protasov, 10--11) In the space, given two intersecting spheres of different radii and a point $ A$ belonging to both spheres. Prove that there is a point $ B$ in the space with the following property:
if an arbitrary circle passes through points $ A$ and $ B$ then the second points of its meet with the given spheres are equidistant from $ B$.
$ ABCD$ is a terahedron: $ AD\plus{}BD\equal{}AC\plus{}BC,$ $ BD\plus{}CD\equal{}BA\plus{}CA,$ $ CD\plus{}AD\equal{}CB\plus{}AB,$ $ M,N,P$ are the mid points of $ BC,CA,AB.$ $ OA\equal{}OB\equal{}OC\equal{}OD.$ Prove that $ \angle MOP \equal{} \angle NOP \equal{}\angle NOM.$
An infinite increasing arithmetical progression consists of positive integers and contains a perfect cube. Prove that this progression also contains a term which is a perfect cube but not a perfect square.
Is it possible to build a $ 4\times 4\times4$ cube from blocks of the following shape consisting of $ 4$ unit cubes?
A block of wood has the shape of a right circular cylinder with radius $6$ and height $8$, and its entire surface has been painted blue. Points $A$ and $B$ are chosen on the edge on one of the circular faces of the cylinder so that $\overarc{AB}$ on that face measures $120^\circ$. The block is then sliced in half along the plane that passes through point $A$, point $B$, and the center of the cylinder, revealing a flat, unpainted face on each half. The area of one of those unpainted faces is $a\cdot\pi + b\sqrt{c}$, where $a$, $b$, and $c$ are integers and $c$ is not divisible by the square of any prime. Find $a+b+c$.
[asy]import three; import solids;
size(8cm);
currentprojection=orthographic(-1,-5,3);
picture lpic, rpic;
size(lpic,5cm);
draw(lpic,surface(revolution((0,0,0),(-3,3*sqrt(3),0)..(0,6,4)..(3,3*sqrt(3),8),Z,0,120)),gray(0.7),nolight);
draw(lpic,surface(revolution((0,0,0),(-3*sqrt(3),-3,8)..(-6,0,4)..(-3*sqrt(3),3,0),Z,0,90)),gray(0.7),nolight);
draw(lpic,surface((3,3*sqrt(3),8)..(-6,0,8)..(3,-3*sqrt(3),8)--cycle),gray(0.7),nolight);
draw(lpic,(3,-3*sqrt(3),8)..(-6,0,8)..(3,3*sqrt(3),8));
draw(lpic,(-3,3*sqrt(3),0)--(-3,-3*sqrt(3),0),dashed);
draw(lpic,(3,3*sqrt(3),8)..(0,6,4)..(-3,3*sqrt(3),0)--(-3,3*sqrt(3),0)..(-3*sqrt(3),3,0)..(-6,0,0),dashed);
draw(lpic,(3,3*sqrt(3),8)--(3,-3*sqrt(3),8)..(0,-6,4)..(-3,-3*sqrt(3),0)--(-3,-3*sqrt(3),0)..(-3*sqrt(3),-3,0)..(-6,0,0));
draw(lpic,(6*cos(atan(-1/5)+3.14159),6*sin(atan(-1/5)+3.14159),0)--(6*cos(atan(-1/5)+3.14159),6*sin(atan(-1/5)+3.14159),8));
size(rpic,5cm);
draw(rpic,surface(revolution((0,0,0),(3,3*sqrt(3),8)..(0,6,4)..(-3,3*sqrt(3),0),Z,230,360)),gray(0.7),nolight);
draw(rpic,surface((-3,3*sqrt(3),0)..(6,0,0)..(-3,-3*sqrt(3),0)--cycle),gray(0.7),nolight);
draw(rpic,surface((-3,3*sqrt(3),0)..(0,6,4)..(3,3*sqrt(3),8)--(3,3*sqrt(3),8)--(3,-3*sqrt(3),8)--(3,-3*sqrt(3),8)..(0,-6,4)..(-3,-3*sqrt(3),0)--cycle),white,nolight);
draw(rpic,(-3,-3*sqrt(3),0)..(-6*cos(atan(-1/5)+3.14159),-6*sin(atan(-1/5)+3.14159),0)..(6,0,0));
draw(rpic,(-6*cos(atan(-1/5)+3.14159),-6*sin(atan(-1/5)+3.14159),0)..(6,0,0)..(-3,3*sqrt(3),0),dashed);
draw(rpic,(3,3*sqrt(3),8)--(3,-3*sqrt(3),8));
draw(rpic,(-3,3*sqrt(3),0)..(0,6,4)..(3,3*sqrt(3),8)--(3,3*sqrt(3),8)..(3*sqrt(3),3,8)..(6,0,8));
draw(rpic,(-3,3*sqrt(3),0)--(-3,-3*sqrt(3),0)..(0,-6,4)..(3,-3*sqrt(3),8)--(3,-3*sqrt(3),8)..(3*sqrt(3),-3,8)..(6,0,8));
draw(rpic,(-6*cos(atan(-1/5)+3.14159),-6*sin(atan(-1/5)+3.14159),0)--(-6*cos(atan(-1/5)+3.14159),-6*sin(atan(-1/5)+3.14159),8));
label(rpic,"$A$",(-3,3*sqrt(3),0),W);
label(rpic,"$B$",(-3,-3*sqrt(3),0),W);
add(lpic.fit(),(0,0));
add(rpic.fit(),(1,0));[/asy]
$ABCD$ is a rectangle with $BC / AB = \sqrt2$. $ABEF$ is a congruent rectangle in a different plane. Find the angle $DAF$ such that the lines $CA$ and $BF$ are perpendicular. In this configuration, find two points on the line $CA$ and two points on the line $BF$ so that the four points form a regular tetrahedron.
Prove that the polynomial $z^{2n} + z^n + 1\ (n \in \mathbb{N})$ is divisible by the polynomial $z^2 + z + 1$ if and only if $n$ is not a multiple of $3$.
Given a plane $P$ in space. For a figure $A$, call orthogonal projection the whole of points of intersection between the perpendicular drawn from each point in $A$ and $P$. Answer the following questions.
(1) Let a plane $Q$ intersects with the plane $P$ by angle $\theta\ \left(0<\theta <\frac{\pi}{2}\right)$ between the planes, that is to say, the angles between two lines, is $\theta$, which can be generated by cuttng $P,\ Q$ by a plane which is perpendicular to the line of intersection of $P$ and $Q$. Find the maximum and minimum length of the orthogonal projection of the line segment in length 1 on $Q$ on to $P$..
(2) Consider $Q$ in (1). Find the area of the orthogonal projection of a equilateral triangle on $Q$ with side length 1 onto $P$.
(3) What's the shape of the orthogonal projection $T'$ of a regular tetrahedron $T$ with side length 1 on to $P'$, then find the max area of $T'$.
Let the edges of rectangular parallelepiped be $x,y$ and $z$ ($x<y<z$). Let
$$p=4(x+y+z), s=2(xy+yz+zx) \,\,\, and \,\,\, d=\sqrt{x^2+y^2+z^2}$$ be its perimeter, surface area and diagonal length, respectively. Prove that $$x < \frac{1}{3}\left( \frac{p}{4}- \sqrt{d^2 - \frac{s}{2}}\right )\,\,\, and \,\,\, z > \frac{1}{3}\left( \frac{p}{4}- \sqrt{d^2 - \frac{s}{2}}\right )$$
In the right trapezoid $ABCD$ with $AB \parallel CD, \angle B = 90^o$ and $AB = 2DC$.
At points $A$ and $D$ there is therefore a part of the plane $(ABC)$ perpendicular to the plane of the trapezoid, on which the points $N$ and $P$ are taken, ($AP$ and $PD$ are perpendicular to the plane) such that $DN = a$ and $AP = \frac{a}{2}$ . Knowing that $M$ is the midpoint of the side $BC$ and the triangle $MNP$ is equilateral, determine:
a) the cosine of the angle between the planes $MNP$ and $ABC$.
b) the distance from $D$ to the plane $MNP$