Found problems: 2265
In space there is a line $ k $ and a cube with a vertex $ M $ and edges $ \overline{MA} $, $ \overline{MB} $, $ \overline{MC} $, of length$ 1$. Prove that the length of the orthogonal projection of edge $ MA $ on the line $ k $ is equal to the area of the orthogonal projection of a square with sides $ MB $ and $ MC $ onto a plane perpendicular to the line $ k $.
[hide=original wording]W przestrzeni dana jest prosta $ k $ oraz sześcian o wierzchołku $ M $ i krawędziach $ \overline{MA} $, $ \overline{MB} $, $ \overline{MC} $, długości 1. Udowodnić, że długość rzutu prostokątnego krawędzi $ MA $ na prostą $ k $ jest równa polu rzutu prostokątnego kwadratu o bokach $ MB $ i $ MC $ na płaszczyznę prostopadłą do prostej $ k $.[/hide]
The altitudes of a tetrahedron meet at a single point. Prove that this point, the centroid of one face of the tetrahedron, the foot of the altitude on that face, and the three points dividing the other three altitudes in ratio $2:1$ (closer to the feet) all lie on a sphere.
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 $1$.
It is given a cube with sidelength $a$. Find the surface of the intersection of the cube with a plane, perpendicular to one of its diagonals and whose distance from the centre of the cube is equal to $h$.
Prove that the product of two natural numbers with their sum cannot be the third power of a natural number.
We consider a cube with sides of length 1. Prove that a tetrahedron with vertices in the set of the vertices of the cube has the volume $\dfrac 16$ if and only if 3 of the vertices of the tetrahedron are vertices on the same face of the cube.
[i]Dinu Serbanescu[/i]
In each lateral face of a pentagonal prism at least one of the four angles is equal to $f$. Find all possible values of $f$.
(A Shapovalov)
$L, L'$ are two skew lines in space and $p$ is a plane not containing either line. $M$ is a variable line parallel to $p$ which meets $L$ at $X$ and $L'$ at $Y$. Find the position of $M$ which minimises the distance $XY$. $L''$ is another fixed line. Find the line $M$ which is also perpendicular to $L''$ .
Find the greatest positive integer $n$ for which $n$ is divisible by all positive integers whose cube is not greater than $n.$
A right circular cone pointing downward forms an angle of $60^\circ$ at its vertex. Sphere $S$ with radius $1$ is set into the cone so that it is tangent to the side of the cone. Three congruent spheres are placed in the cone on top of S so that they are all tangent to each other, to sphere $S$, and to the side of the cone. The radius of these congruent spheres can be written as $\tfrac{a+\sqrt{b}}{c}$ where $a$, $b$, and $c$ are positive integers such that $a$ and $c$ are relatively prime. Find $a + b + c$.
[asy]
size(150);
real t=0.12;
void ball(pair x, real r, real h, bool ww=true)
{
pair xx=yscale(t)*x+(0,h);
path P=circle(xx,r);
unfill(P);
draw(P);
if(ww) draw(ellipse(xx-(0,r/2),0.85*r,t*r));
}
pair X=(0,0);
real H=17, h=5, R=h/2;
draw(H*dir(120)--(0,0)--H*dir(60));
draw(ellipse((0,0.87*H),H/2,t*H/2));
pair Y=(R,h+2*R),C=(0,h);
real r;
for(int k=0;k<20;++k)
{
r=-(dir(30)*Y).x;
Y-=(sqrt(3)/2*Y.x-r,abs(Y-C)-R-r)/3;
}
ball(Y.x*dir(90),r,Y.y,false);
ball(X,R,h);
ball(Y.x*dir(-30),r,Y.y);
ball(Y.x*dir(210),r,Y.y);[/asy]
A starship is located in a halfspace at the distance $a$ from its boundary. The crew knows this but does not know which direction to move to reach the boundary plane. The starship may travel through the space by any path, may measure the way it has already travelled and has a sensor that signals when the boundary is reached. Is it possible to reach the boundary for sure, having passed no more than:
$a)14a$
$b)13a$?
There are $27$ unit cubes. We are marking one point on each of the two opposing faces, two points on each of the other two opposing faces, and three points on each of the remaining two opposing faces of each cube. We are constructing a $3\times 3 \times 3$ cube with these $27$ cubes. What is the least number of marked points on the faces of the new cube?
$
\textbf{(A)}\ 54
\qquad\textbf{(B)}\ 60
\qquad\textbf{(C)}\ 72
\qquad\textbf{(D)}\ 90
\qquad\textbf{(E)}\ 96
$
a) Let $S$ and $P$ be the area and perimeter of some triangle. The straight lines on which its sides are located move to the outside by a distance $h$. What will be the area and perimeter of the triangle formed by the three obtained lines?
b) Let $V$ and $S$ be the volume and surface area of some tetrahedron. The planes on which its faces are located are moved to the outside by a distance $h$. What will be the volume and surface area of the tetrahedron formed by the three obtained planes?
Is it possible to number the $8$ vertices of a cube from $1$ to $8$ in such a way that the value of the sum on every edge is different?
A vertex of a tetrahedron is called perfect if the three edges at this vertex are sides of a certain triangle. How many perfect vertices can a tetrahedron have?
A plane cuts a right circular cone of volume $ V$ into two parts. The plane is tangent to the circumference of the base of the cone and passes through the midpoint of the altitude. Find the volume of the smaller part.
[i]Original formulation:[/i]
A plane cuts a right circular cone into two parts. The plane is tangent to the circumference of the base of the cone and passes through the midpoint of the altitude. Find the ratio of the volume of the smaller part to the volume of the whole cone.
Let there be given lines $a,b,c$ in the space, no two of which are parallel. Suppose that there exist planes $\alpha,\beta,\gamma$ which contain $a,b,c$ respectively, which are perpendicular to each other. Construct the intersection point of these three planes. (A space construction permits drawing lines, planes and spheres and translating objects for any vector.)
We have a polyhedron such that an ant can walk from one vertex to another, traveling only along edges, and traversing every edge exactly once. What is the smallest possible total number of vertices, edges, and faces of this polyhedron?
Three of the edges of a cube are $\overline{AB}, \overline{BC},$ and $\overline{CD},$ and $\overline{AD}$ is an interior diagonal. Points $P, Q,$ and $R$ are on $\overline{AB}, \overline{BC},$ and $\overline{CD},$ respectively, so that $AP=5, PB=15, BQ=15,$ and $CR=10.$ What is the area of the polygon that is the intersection of plane $PQR$ and the cube?
Two real numbers are selected independently at random from the interval [-20, 10]. What is the probability that the product of those numbers is greater than zero?
$ \textbf{(A)}\ \frac{1}{9} \qquad
\textbf{(B)}\ \frac{1}{3} \qquad
\textbf{(C)}\ \frac{4}{9} \qquad
\textbf{(D)}\ \frac{5}{9} \qquad
\textbf{(E)}\ \frac{2}{3} $
The bottom, side, and front areas of a rectangular box are known. The product of these areas is equal to:
$ \textbf{(A)}\ \text{the volume of the box} \qquad\textbf{(B)}\ \text{the square root of the volume} \qquad\textbf{(C)}\ \text{twice the volume}$
$ \textbf{(D)}\ \text{the square of the volume} \qquad\textbf{(E)}\ \text{the cube of the volume}$
Prove that we can put $ \Omega(\frac1{\epsilon})$ points on surface of a sphere with radius 1 such that distance of each of these points and the plane passing through center and two of other points is at least $ \epsilon$.
A sphere is inscribed in polyhedral $P$. The faces of $P$ are coloured with black and white in a way that no two black faces share an edge.
Prove that the sum of surface of black faces is less than or equal to the sum of the surface of the white faces.
Time allowed for this problem was 1 hour.
Let $S$ be the set of $n$-uples $\left( x_{1}, x_{2}, \ldots, x_{n}\right)$ such that $x_{i}\in \{ 0, 1 \}$ for all $i \in \overline{1,n}$, where $n \geq 3$. Let $M(n)$ be the smallest integer with the property that any subset of $S$ with at least $M(n)$ elements contains at least three $n$-uples \[\left( x_{1}, \ldots, x_{n}\right), \, \left( y_{1}, \ldots, y_{n}\right), \, \left( z_{1}, \ldots, z_{n}\right) \] such that
\[\sum_{i=1}^{n}\left( x_{i}-y_{i}\right)^{2}= \sum_{i=1}^{n}\left( y_{i}-z_{i}\right)^{2}= \sum_{i=1}^{n}\left( z_{i}-x_{i}\right)^{2}. \]
(a) Prove that $M(n) \leq \left\lfloor \frac{2^{n+1}}{n}\right\rfloor+1$.
(b) Compute $M(3)$ and $M(4)$.
The circular base of a hemisphere of radius $2$ rests on the base of a square pyramid of height $6$. The hemisphere is tangent to the other four faces of the pyramid. What is the edge-length of the base of the pyramid?
$ \textbf{(A)}\ 3\sqrt{2} \qquad
\textbf{(B)}\ \frac{13}{3} \qquad
\textbf{(C)}\ 4\sqrt{2} \qquad
\textbf{(D)}\ 6 \qquad
\textbf{(E)}\ \frac{13}{2}
$