Which of the following does not divide the number of ordered pairs $(x,y)$ of integers satisfying the equation $x^3 - 13y^3 = 1453$? $ \textbf{a)}\ 2 \qquad\textbf{b)}\ 3 \qquad\textbf{c)}\ 5 \qquad\textbf{d)}\ 7 \qquad\textbf{e)}\ \text{None of above} $

A right circular cone has a base with radius 600 and height $200\sqrt{7}$. A fly starts at a point on the surface of the cone whose distance from the vertex of the cone is 125, and crawls along the surface of the cone to a point on the exact opposite side of the cone whose distance from the vertex is $375\sqrt{2}$. Find the least distance that the fly could have crawled.

Given the tetrahedron $ABCD$ whose edges $AB$ and $CD$ have lengths $a$ and $b$ respectively. The distance between the skew lines $AB$ and $CD$ is $d$, and the angle between them is $\omega$. Tetrahedron $ABCD$ is divided into two solids by plane $\epsilon$, parallel to lines $AB$ and $CD$. The ratio of the distances of $\epsilon$ from $AB$ and $CD$ is equal to $k$. Compute the ratio of the volumes of the two solids obtained.

Jesse cuts a circular paper disk of radius $12$ along two radii to form two sectors, the smaller having a central angle of $120$ degrees. He makes two circular cones, using each sector to form the lateral surface of a cone. What is the ratio of the volume of the smaller cone to that of the larger? $ \textbf{(A)}\ \frac{1}{8} \qquad\textbf{(B)}\ \frac{1}{4} \qquad\textbf{(C)}\ \frac{\sqrt{10}}{10} \qquad\textbf{(D)}\ \frac{\sqrt{5}}{6} \qquad\textbf{(E)}\ \frac{\sqrt{10}}{5} $

Let $ABC$ be an equilateral triangle. On the plane $(ABC)$ rise the perpendiculars $AA'$ and $BB'$ on the same side of the plane, so that $AA' = AB$ and $BB' =\frac12 AB$. Determine the measure the angle between the planes $(ABC)$ and $(A'B'C')$.

In space there are given four distinct points $ A $, $ B $, $ C $, $ D $. Prove that the three segments connecting the midpoints of the segments $ AB $ and $ CD $, $ AC $ and $ BD $, $ AD $ and $ BC $ have a common midpoint.

In the rectangular parallelepiped $ABCDA'B'C'D'$, the points $E$ and $F$ are the centers of the faces $ABCD$ and $ADD' A'$, respectively, and the planes $(BCF)$ and $(B'C'E)$ are perpendicular. Let $A'M \perp B'A$, $M \in B'A$ and $BN \perp B'C$, $N \in B'C$. Denote $n = \frac{C'D}{BN}$. a) Show that $n \ge \sqrt2$. . b) Express and in terms of $n$, the ratio between the volume of the tetrahedron $BB'M N$ and the volume of the parallelepiped $ABCDA'B'C'D'$.

Show that if an infinite arithmetic progression of positive integers contains a square and a cube, it must contain a sixth power.

Which of all planes, the one and the same body diagonal of a cube with the edge length $a$, cuts out a cut figure with the smallest area from the cube? Calculate the area of such a cut figure. [hide=original wording]Welche von allen Ebenen, die eine und dieselbe Korperdiagonale eines Wurfels mit der Kantenlange a enthalten, schneiden aus den W¨urfel eine Schnittfigur kleinsten Flacheninhaltes heraus? Berechnen Sie den Fl¨acheninhalt solch einer Schnittfigur![/hide]

Through the centre of a cube (tetrahedron, icosahedron) draw a straight line in such a way that the sum of the squares of its distances from the vertices is a) minimal, b) maximal.

Point $A$ and segment $BC$ are given. Determine the locus of points in space which are vertices of right angles with one side passing through $A$, and the other side intersecting segment $BC$.

4. Consider spherical surface $S$ which radius is $1$, central point $(0,0,1)$ in $xyz$ space. If point $Q$ move to points on S expect $(0,0,2)$. Let $R$ be an intersection of plane $z=0$ and line $l$ pass point $Q$ and point $P (1,0,2)$. Find the range of moving $R$, then illustrate it.

Inside the trihedral angle $ OABC $, whose plane angles $ AOB $, $ BOC $, $ COA $ are equal, a point $ S $ is chosen equidistant from the faces of this angle. Through point $ S $ a plane is drawn that intersects the edges $ OA $, $ OB $, $ OC $ at points $ M $, $ N $, $ P $, respectively. Prove that the sum $$ \frac{1}{OM} + \frac{1}{ON} + \frac{1}{OP}$$ has a constant value, i.e. independent of the position of the plane $ MNP $.

Consider a system of infinitely many spheres made of metal, with centers at points $(a, b, c) \in \mathbb Z^3$. We say that the system is stable if the temperature of each sphere equals the average temperature of the six closest spheres. Assuming that all spheres in a stable system have temperatures between $0^\circ C$ and $1^\circ C$, prove that all the spheres have the same temperature.

Prove that there is no inclined plane such that any tetrahedron placed arbitrarily with a certain face on the plane will not fall over. It means the following: Given a plane $ \pi $ and a line $ l $ not perpendicular to it. Prove that there is a tetrahedron $ T $ such that for each of its faces $ S $ there is in the plane $ \pi $ a triangle $ ABC $ congruent to $ S $ and there is a point $ D $ such that the tetrahedron $ ABCD $ congruent to $ T $ and the line parallel to $ l $ passing through the center of gravity of the tetrahedron $ ABCD $ does not intersect the triangle $ ABC $. Note. The center of gravity of a tetrahedron is the intersection point of the segments connecting the centers of gravity of the faces of this tetrahedron with the opposite vertices (it is known that such a point always exists).

A cube whose edge is $1$ is intersected by a plane that does not pass through any of its vertices, and its edges intersect only at points that are the midpoints of these edges. Find the area of the formed section. Consider all possible cases. (Alexander Shkolny)

Consider the cube $ABCDA'B'C'D'$ (with face $ABCD$ directly above face $A'B'C'D'$). a) Find the locus of the midpoints of the segments $XY$, where $X$ is any point of $AC$ and $Y$ is any piont of $B'D'$; b) Find the locus of points $Z$ which lie on the segment $XY$ of part a) with $ZY=2XZ$.

Calculate number of the hamiltonian cycles of the graph below: (15 points)

You have three colors $\{\text{red}, \text{blue}, \text{green}\}$ with which you can color the faces of a regular octahedron (8 triangle sided polyhedron, which is two square based pyramids stuck together at their base), but you must do so in a way that avoids coloring adjacent pieces with the same color. How many different coloring schemes are possible? (Two coloring schemes are considered equivalent if one can be rotated to fit the other.)

In the tetrahedron $SABC$, points $E, F, K, L$ are the midpoints of the sides $SA , BC, AC, SB$ respectively, . The lengths of the segments $EF$ and $KL$ are equal to $11 cm$ and $13 cm$ respectively, and the length of the segment $AB$ equals to $18 cm$. Find the length of the side $SC$ of the tetrahedron.

An ant crawls along the edges of a cube turning only at its vertices. It has visited one of the vertices $25$ times. Is it possible that it has visited each of the other $7$ vertices exactly $20$ times? (S Tokarev)

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 $a+b+c$.

The altitudes of a tetrahedron intersect in a point. Prove that this point, the foot of one of the altitudes, and the points dividing the other three altitudes in the ratio $2 : 1$ (measuring from the vertices) lie on a sphere. [i]D. Tereshin[/i]

The region consisting of all points in three-dimensional space within $3$ units of line segment $\overline{AB}$ has volume $216\pi$. What is the length $AB$? $\textbf{(A)}\ 6\qquad\textbf{(B)}\ 12\qquad\textbf{(C)}\ 18\qquad\textbf{(D)}\ 20\qquad\textbf{(E)}\ 24$

For a given integer $n\geq2$, a pyramid of height $n$ if defined as a collection of $1^2+2^2+\dots+n^2$ stone cubes of equal size stacked in $n$ layers such that the cubes in the $k$-th layer form a square with sidelength $n+1-k$ and every cube (except for the ones in the bottom layer) rests on four cubes in the layer below. Some of the cubes are made of sandstone, some are made of granite. The top cube is made of granite, and to ensure the stability of the piramid, for each granite cube (except for the ones in the bottom layer), at least three out of four of the cubes supporting it have to be granite. What is the minimum possible number of granite cubes in such an arrangement?