What is the largest number of parts into which $n$ planes can divide space? We assume that the set of planes is non-degenerate in the sense that any three planes intersect in one point and no four planes have a common point (and for n=2 it is necessary to require that the planes are not parallel).

Let $ R$ denote a non-negative rational number. Determine a fixed set of integers $ a,b,c,d,e,f$, such that for [i]every[/i] choice of $ R$, \[ \left| \frac{aR^2\plus{}bR\plus{}c}{dR^2\plus{}eR\plus{}f}\minus{}\sqrt[3]{2}\right| < \left|R\minus{}\sqrt[3]{2}\right|.\]

A quadrilateral pyramid is cut by a plane parallel to the base. Suppose that a sphere $S$ is circumscribed and a sphere $\Sigma$ inscribed in the obtained solid, and moreover that the line through the centers of these two spheres is perpendicular to the base of the pyramid. Show that the pyramid is regular.

Faces of a convex polyhedron are six squares and 8 equilateral triangles and each edge is a common side for one triangle and one square. All dihedral angles obtained from the triangle and square with a common edge, are equal. Prove that it is possible to circumscribe a sphere around the polyhedron, and compute the ratio of the squares of volumes of that polyhedron and of the ball whose boundary is the circumscribed sphere.

We glue together $990$ one inch cubes into a $9$ by $10$ by $11$ inch rectangular solid. Then we paint the outside of the solid. How many of the original $990$ cubes have just one of their sides painted?

An ice-cream novelty item consists of a cup in the shape of a $4$-inch-tall frustum of a right circular cone, with a $2$-inch-diameter base at the bottom and a $4$-inch-diameter base at the top, packed solid with ice cream, together with a solid cone of ice cream of height $4$ inches, whose base, at the bottom, is the top base of the frustum. What is the total volume of the ice cream, in cubic inches? $\textbf{(A)}\ 8\pi\qquad\textbf{(B)}\ \frac{28\pi}{3}\qquad\textbf{(C)}\ 12\pi\qquad\textbf{(D)}\ 14\pi\qquad\textbf{(E)}\ \frac{44\pi}{3}$

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.

In the tetrahedron $ABCD$, $E$ and $F$ are the midpoints of $BC$ and $AD$, $G$ is the midpoint of the segment $EF$. Construct a plane through $G$ intersecting the segments $AB$, $AC$, $AD$ in the points $M,N,P$ respectively in such a way that the sum of the volumes of the tetrahedrons $BMNP$, $CMNP$ and $DMNP$ to be minimal. [i]H. Lesov[/i]

A sphere with center at $O$ is inscribed in a trihedral angle with vertex $S$. Prove that the plane passing through the three tangent points is perpendicular to $OS$.

The volume of a cube (in cubic inches) plus three times the total length of its edges (in inches) is equal to twice its surface area (in square inches). How many inches long is its long diagonal?

A shape is created by joining seven unit cubes, as shown. What is the ratio of the volume in cubic units to the surface area in square units? [asy] import three; defaultpen(linewidth(0.8)); real r=0.5; currentprojection=orthographic(1,1/2,1/4); draw(unitcube, white, thick(), nolight); draw(shift(1,0,0)*unitcube, white, thick(), nolight); draw(shift(1,-1,0)*unitcube, white, thick(), nolight); draw(shift(1,0,-1)*unitcube, white, thick(), nolight); draw(shift(2,0,0)*unitcube, white, thick(), nolight); draw(shift(1,1,0)*unitcube, white, thick(), nolight); draw(shift(1,0,1)*unitcube, white, thick(), nolight);[/asy] $\textbf{(A)} \:1 : 6 \qquad\textbf{ (B)}\: 7 : 36 \qquad\textbf{(C)}\: 1 : 5 \qquad\textbf{(D)}\: 7 : 30\qquad\textbf{ (E)}\: 6 : 25$

In cube $ABCD-A_1B_1C_1D_1$, the degree of dihedral angle $A-BD_1-A_1$ is________.

Consider all tetrahedra $ABCD$ in which the sum of the areas of the faces $ABD, ACD, BCD$ does not exceed $1$. Among such tetrahedra, find those with the maximum volume.

In a country, towns are connected by roads. Each town is directly connected to exactly three other towns. Show that there exists a town from which you can make a round-trip, without using the same road more than once, and for which the number of roads used is not divisible by $3$. (Not all towns need to be visited.)

The edges of a cube are labeled from 1 to 12 in an arbitrary manner. Show that it is not possible to get the sum of the edges at each vertex the same. Show that we can get eight vertices with the same sum if one of the labels is changed to 13.

A quadrilateral pyramid and a regular tetrahedron have edges that are all equal in length. They are glued together so that they have in common $1$ equilateral triangle . Prove that the resulting body has exactly $5$ sides.

A wooden beam $EFGH$ $ABCD$ is with three cuts in $8$ smaller ones sawn beams. Each cut is parallel to one of the three pair of opposit sides. Each pair of saw cuts is shown perpendicular to each other. The smaller bars at the corners $A, C, F$ and $H$ have a capacity of $9, 12, 8, 24$ respectively.(The proportions in the picture are not correct!!). Calculate content of the entire bar. [asy] unitsize (0.5 cm); pair A, B, C, D, E, F, G, H; pair x, y, z; x = (1,0.5); y = (-0.8,0.8); z = (0,1); B = (0,0); C = 5*x; A = 3*y; F = 4*z; E = A + F; G = C + F; H = A + C + F; fill(y--3*y--(3*y + z)--(y + z)--cycle, gray(0.8)); fill(2*x--5*x--(5*x + z)--(2*x + z)--cycle, gray(0.8)); fill((y + z)--(y + 4*z)--(y + 4*z + 2*x)--(4*z + 2*x)--(2*x + z)--z--cycle, gray(0.8)); fill((2*x + y + 4*z)--(2*x + 3*y + 4*z)--(5*x + 3*y + 4*z)--(5*x + y + 4*z)--cycle, gray(0.8)); draw(B--C--G--H--E--A--cycle); draw(B--F); draw(E--F); draw(G--F); draw(y--(y + 4*z)--(y + 4*z + 5*x)); draw(2*x--(2*x + 4*z)--(2*x + 4*z + 3*y)); draw((3*y + z)--z--(5*x + z)); label("$A$", A, SW); label("$B$", B, S); label("$C$", C, SE); label("$E$", E, NW); label("$F$", F, SE); label("$G$", G, NE); label("$H$", H, N); [/asy]

A tetrahedron $ABCD$ is divided into five polyhedra so that each face of the tetrahedron is a face of (exactly) one polyhedron, and that the intersection of any two of the polyhedra is either a common vertex, a common edge, or a common face. What is the smallest possible sum of the numbers of faces of the five polyhedra?

Two circles with radii 1 meet in points $X, Y$, and the distance between these points also is equal to $1$. Point $C$ lies on the first circle, and lines $CA, CB$ are tangents to the second one. These tangents meet the first circle for the second time in points $B', A'$. Lines $AA'$ and $BB'$ meet in point $Z$. Find angle $XZY$.

We call the set $A\in \mathbb R^n$ CN if and only if for every continuous $f:A\to A$ there exists some $x\in A$ such that $f(x)=x$. a) Example: We know that $A = \{ x\in\mathbb R^n | |x|\leq 1 \}$ is CN. b) The circle is not CN. Which one of these sets are CN? 1) $A=\{x\in\mathbb R^3| |x|=1\}$ 2) The cross $\{(x,y)\in\mathbb R^2|xy=0,\ |x|+|y|\leq1\}$ 3) Graph of the function $f:[0,1]\to \mathbb R$ defined by \[f(x)=\sin\frac 1x\ \mbox{if}\ x\neq0,\ f(0)=0\]

Let $MABCD$ be a pyramid with the square $ABCD$ as the base, in which $MA=MD$, $MA^2+AB^2=MB^2$ and the area of $\triangle ADM$ is equal to $1$. Determine the radius of the largest ball that is contained in the given pyramid.

A three-quarter sector of a circle of radius $4$ inches together with its interior can be rolled up to form the lateral surface area of a right circular cone by taping together along the two radii shown. What is the volume of the cone in cubic inches? [asy] draw(Arc((0,0), 4, 0, 270)); draw((0,-4)--(0,0)--(4,0)); label("$4$", (2,0), S); [/asy] $\textbf{(A)}\ 3\pi \sqrt5 \qquad\textbf{(B)}\ 4\pi \sqrt3 \qquad\textbf{(C)}\ 3 \pi \sqrt7 \qquad\textbf{(D)}\ 6\pi \sqrt3 \qquad\textbf{(E)}\ 6\pi \sqrt7$

If all edges of a non-planar quadrilateral tangent the faces of a sphere, prove that all of the points of tangency belong to a plane.

Find all integers $n \ge 3$ with the following property: it is possible to assign pairwise different positive integers to the vertices of an $n$-gonal prism in such a way that vertices with labels $a$ and $b$ are connected by an edge if and only if $a | b$ or $b | a$. Poland

Find all the positive integers less than 1000 such that the cube of the sum of its digits is equal to the square of such integer.