One dimension of a cube is increased by $ 1$, another is decreased by $ 1$, and the third is left unchanged. The volume of the new rectangular solid is $ 5$ less than that of the cube. What was the volume of the cube? $ \textbf{(A)}\ 8 \qquad \textbf{(B)}\ 27 \qquad \textbf{(C)}\ 64 \qquad \textbf{(D)}\ 125 \qquad \textbf{(E)}\ 216$

Find the volume of the uion $ A\cup B\cup C$ of the three subsets $ A,\ B,\ C$ in $ xyz$ space such that: \[ A\equal{}\{(x,\ y,\ z)\ |\ |x|\leq 1,\ y^2\plus{}z^2\leq 1\}\] \[ B\equal{}\{(x,\ y,\ z)\ |\ |y|\leq 1,\ z^2\plus{}x^2\leq 1\}\] \[ C\equal{}\{(x,\ y,\ z)\ |\ |z|\leq 1,\ x^2\plus{}y^2\leq 1\}\]

An ant is on one face of a cube. At every step, the ant walks to one of its four neighboring faces with equal probability. What is the expected (average) number of steps for it to reach the face opposite its starting face?

Prove that a tetrahedron with just one edge length greater than $1$ has volume at most $ \frac{1}{8}.$

In a tetrahedron, segments connecting the midpoints of heights with the orthocenters of the faces to which these heights are drawn intersect at one point. Prove that in such a tetrahedron all faces are equal or there are perpendicular edges. (Yu. Blinkov)

A $n \times n \times n$ cube is constructed using $1 \times 1 \times 1$ cubes, some of them black and others white, such that in each $n \times 1 \times 1$, $1 \times n \times 1$, and $1 \times 1 \times n$ subprism there are exactly two black cubes, and they are separated by an even number of white cubes (possibly 0). Show it is possible to replace half of the black cubes with white cubes such that each $n \times 1 \times 1$, $1 \times n \times 1$ and $1 \times 1 \times n$ subprism contains exactly one black cube.

How many unordered pairs of edges of a given cube determine a plane? $\textbf{(A) } 21 \qquad\textbf{(B) } 28 \qquad\textbf{(C) } 36 \qquad\textbf{(D) } 42 \qquad\textbf{(E) } 66$

Let be given a tetrahedron $ ABCD$ such that triangle $ BCD$ equilateral and $ AB \equal{} AC \equal{} AD$. The height is $ h$ and the angle between two planes $ ABC$ and $ BCD$ is $ \alpha$. The point $ X$ is taken on $ AB$ such that the plane $ XCD$ is perpendicular to $ AB$. Find the volume of the tetrahedron $ XBCD$.

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] $\textbf{(A)} \:1 : 6 \qquad\textbf{ (B)}\: 7 : 36 \qquad\textbf{(C)}\: 1 : 5 \qquad\textbf{(D)}\: 7 : 30\qquad\textbf{ (E)}\: 6 : 25$

A unit cube is cut twice to form three triangular prisms, two of which are congruent, as shown in Figure 1. The cube is then cut in the same manner along the dashed lines shown in Figure 2. This creates nine pieces. What is the volume of the piece that contains vertex $ W$? [asy]import three; size(200); defaultpen(linewidth(.8pt)+fontsize(10pt)); currentprojection=oblique; path3 p1=(0,2,2)--(0,2,0)--(2,2,0)--(2,2,2)--(0,2,2)--(0,0,2)--(2,0,2)--(2,2,2); path3 p2=(2,2,0)--(2,0,0)--(2,0,2); path3 p3=(0,0,2)--(0,2,1)--(2,2,1)--(2,0,2); path3 p4=(2,2,1)--(2,0,0); pen finedashed=linetype("4 4"); draw(p1^^p2^^p3^^p4); draw(shift((4,0,0))*p1); draw(shift((4,0,0))*p2); draw(shift((4,0,0))*p3); draw(shift((4,0,0))*p4); draw((4,0,2)--(5,2,2)--(6,0,2),finedashed); draw((5,2,2)--(5,2,0)--(6,0,0),finedashed); label("$W$",(3,0,2)); draw((2.7,.3,2)--(2.1,1.9,2),linewidth(.6pt)); draw((3.4,.3,2)--(5.9,1.9,2),linewidth(.6pt)); label("Figure 1",(1,-0.5,2)); label("Figure 2",(5,-0.5,2));[/asy]$ \textbf{(A)}\ \frac {1}{12}\qquad \textbf{(B)}\ \frac {1}{9}\qquad \textbf{(C)}\ \frac {1}{8}\qquad \textbf{(D)}\ \frac {1}{6}\qquad \textbf{(E)}\ \frac {1}{4}$

Prove the following assertion: The four altitudes of a tetrahedron $ABCD$ intersect in a point if and only if \[AB^2 + CD^2 = BC^2 + AD^2 = CA^2 + BD^2.\]

The sum of the squares of three positive numbers is $160$. One of the numbers is equal to the sum of the other two. The difference between the smaller two numbers is $4.$ What is the difference between the cubes of the smaller two numbers? [i]Author: Ray Li[/i] [hide="Clarification"]The problem should ask for the positive difference.[/hide]

$8$ ants are placed on the edges of the unit cube. Prove that there exists a pair of ants at a distance not exceeding $1$.

Find all natural numbers $n$ for which there exists a convex polyhedron satisfying the following conditions: (i) Each face is a regular polygon. (ii) Among the faces, there are polygons with at most two different numbers of edges. (iii) There are two faces with common edge that are both $n$-gons.

Let $T$ be an arbitrary tetrahedron satisfying the following conditions: [b]I.[/b] Each its side has length not greater than 1, [b]II.[/b] Each of its faces is a right triangle. Let $s(T) = S^2_{ABC} + S^2_{BCD} + S^2_{CDA} + S^2_{DAB}$. Find the maximal possible value of $s(T)$.