Found problems: 2265
If you multiply the number of faces that a pyramid has with the number of edges of the pyramid, you get $5.100$. Determine the number of faces of the pyramid.
$N^3$ unit cubes are made into beads by drilling a hole through them along a diagonal, put on a string and binded. Thus the cubes can move freely in space as long as the vertices of two neighboring cubes (including the first and last one) are touching. For which $N$ is it possible to build a cube of edge $N$ using these cubes?
Consider cubes of edge length 5 composed of 125 cubes of edge length 1 where each of the 125 cubes is either coloured black or white. A cube of edge length 5 is called "big", a cube od edge length is called "small". A posititve integer $ n$ is called "representable" if there is a big cube with exactly $ n$ small cubes where each row of five small cubes has an even number of black cubes whose centres lie on a line with distances $ 1,2,3,4$ (zero counts as even number).
(a) What is the smallest and biggest representable number?
(b) Construct 45 representable numbers.
In regular triangular pyramid $S-ABC$, hieght $SO=3$, length of sides of bottom surface is $6$. Projection of $A$ on plane $SBC$ is $O'$. $P\in AO',\frac{AP}{PO'}=8$. Draw a plane
parallel to plane $ABC$ and passes $P$. Find the area of the cross section.
Answer wither (i) or (ii):
(i) Let $V, V_1 , V_2$ and $V_3$ denote four vertices of a cube such that $V_1 , V_2 , V_3 $ are adjacent to $V.$ Project the cube orthogonally on a plane of which the points are marked with complex numbers. Let the projection of $V$ fall in the origin and the projections of $V_1 , V_2 , V_3 $ in points marked with the complex numbers $z_1 , z_2 , z_3$, respectively. Show that $z_{1}^{2} +z_{2}^{2} +z_{3}^{2}=0.$
(ii) Let $(a_{ij})$ be a matrix such that
$$|a_{ii}| > |a_{i1}| + |a_{i2}|+\ldots +|a_{i i-1}|+ |a_{i i+1}| +\ldots +|a_{in}|$$
for all $i.$ Show that the determinant is not equal to $0.$
Let $S=\{(x,y,z)\in \mathbb{Z}^3\}$ the set of points with integer coordinates in the space. Gugu has infinitely many solid spheres. All with radii $\ge (\frac{\pi}2)^3$. Is it possible for Gugu to cover all points of $S$ with his spheres?
A semicircle is joined to the side of a triangle, with the common edge removed. Sixteen points are arranged on the figure, as shown below. How many non-degenerate triangles can be drawn from the given points?
[asy]
draw((0,-2)--arc((0,0),1,0,180)--cycle);
dot((-0.8775,-0.245));
dot((-0.735,-0.53));
dot((-0.5305,-0.939));
dot((-0.3875,-1.225));
dot((-0.2365,-1.527));
dot((0.155,-1.69));
dot((0.306,-1.388));
dot((0.4,-1.2));
dot((0.551,-0.898));
dot((0.837,-0.326));
dot(dir(25));
dot(dir(50));
dot(dir(65));
dot(dir(100));
dot(dir(115));
dot(dir(140));
[/asy]
In a tetrahedron, the midpoints of all the edges lie on the same sphere. Prove that it's altitudes intersect at one point.
It is given a tetrahedron $ABCD$ and a plane $\alpha$ intersecting the three edges passing through $D$. Prove that $\alpha$ divides the surface of the tetrahedron into two parts proportional to the volumes of the bodies formed if and only if $\alpha$ is passing through the center of the inscribed tetrahedron sphere.
A tank has the shape of a regular hexagonal prism, whose bases are $1$ m on a side and its height is $10$ m. The lateral edges are placed in an oblique position and is partially filled with $9$ m$^3$ of water. The plane of the free surface of the water cuts to all lateral edges. One of them is left with a part of $2$ m under water. What part is under water on the opposite side edge of the prism?
A right cone in $xyz$-space has its apex at $(0,0,0)$, and the endpoints of a diameter on its base are $(12,13,-9)$ and $(12,-5,15)$. The volume of the cone can be expressed as $a\pi$. What is $a$?
Let $E$ be a finite set of points such that $E$ is not contained in a plane and no three points of $E$ are collinear. Show that at least one of the following alternatives holds:
(i) $E$ contains five points that are vertices of a convex pyramid having no other points in common with $E;$
(ii) some plane contains exactly three points from $E.$
Suppose that $X$ is an arbitrary point inside a tetrahedron. Through each vertex of the tetrahedron, draw a straight line that is parallel to the line segment connecting $X$ with the intersection point of the medians of the opposite face. Prove that these four lines meet at the same point.
Two boundary points of a ball of radius 1 are joined by a curve contained in the ball and having length less than 2. Prove that the curve is contained entirely within some hemisphere of the given ball.
How many numbers among $1,2,\ldots,2012$ have a positive divisor that is a cube other than $1$?
$\text{(A) }346\qquad\text{(B) }336\qquad\text{(C) }347\qquad\text{(D) }251\qquad\text{(E) }393$
Let $\vec{v_1},\vec{v_2},\vec{v_3},\vec{v_4}$ and $\vec{v_5}$ be vectors in three dimensions. Show that for some $i,j$ in $1,2,3,4,5$, $\vec{v_i}\cdot \vec{v_j}\ge 0$.
When a right triangle is rotated about one leg, the volume of the cone produced is $800 \pi$ $\text{cm}^3$. When the triangle is rotated about the other leg, the volume of the cone produced is $1920 \pi$ $\text{cm}^3$. What is the length (in cm) of the hypotenuse of the triangle?
Let $AB,AC$ and $AD$ be the edges of a cube, $AB=\alpha$. Point $E$ was marked on the ray $AC$ so that $AE=\lambda \alpha$, and point $F$ was marked on the ray $AD$ so that $AF=\mu \alpha$ ($\mu> 0, \lambda >0$). Find (characterize) pairs of numbers $\lambda$ and $\mu$ such that the cross-sectional area of a cube by any plane parallel to the plane $BCD$ is equal to the cross-sectional area of the tetrahedron $ABEF$ by the same plane.
Show that for any tetrahedron, the condition that opposite edges have the same length is equivalent to the condition that the segment drawn between the midpoints of any two opposite edges is perpendicular to the two edges.
Given a (not necessarily regular) tetrahedron, all of its sides are equal in area. Prove that the following points then coincide:
a) the center of the inscribed sphere, i.e. all four side surfaces internally touching sphere,
b) the center of the surrounding sphere, i.e. the sphere passing through the four vertixes.
Let a right cone of the base radius $r=3$ and height greater than $6$ be inscribed in a sphere of radius $R=6$. The volume of the cone can be expressed as $\pi(a\sqrt{b}+c)$, where $b$ is square free. Find $a+b+c$.
A cube $ABCDA_1B_1C_1D_1$ is given. Find the locus of points $E$ on the face $A_1B_1C_1D_1$ for which there exists a line intersecting the lines $AB$, $A_1D_1$, $B_1D$, and $EC$.
In a tetrahedron $ ABCD $ the edges $AD$, $ BD $, $ CD $ are equal. $ ABC $ Non-collinear points are chosen in the plane. $ A_1$, $B_1$, $C_1 $ The lines $DA_1$, $DB_1$, $DC_1 $ intersect the surface of the sphere circumscribed about the tetrahedron at points $ A_2$, $B_2$, $C_2 $, different from the point $ D $. Prove that the points $A_1$, $B_1$, $C_1$, $A_2$, $B_2$, $C_2$ lie on the surface of a certain sphere.
In the interior of a cube we consider $\displaystyle 2003$ points. Prove that one can divide the cube in more than $\displaystyle 2003^3$ cubes such that any point lies in the interior of one of the small cubes and not on the faces.
Prove that any rigid flat triangle $T$ of area less than $4$ can be inserted through a triangular hole $Q$ with area $3$.