The vertices of a prism are colored using two colors, so that each lateral edge has its vertices differently colored. Consider all the segments that join vertices of the prism and are not lateral edges. Prove that the number of such segments with endpoints differently colored is equal to the number of such segments with endpoints of the same color.

Given four points $ A_1, A_2, A_3, A_4$ in the plane, no three collinear, such that \[ A_1A_2 \cdot A_3 A_4 \equal{} A_1 A_3 \cdot A_2 A_4 \equal{} A_1 A_4 \cdot A_2 A_3, \] denote by $ O_i$ the circumcenter of $ \triangle A_j A_k A_l$ with $ \{i,j,k,l\} \equal{} \{1,2,3,4\}.$ Assuming $ \forall i A_i \neq O_i ,$ prove that the four lines $ A_iO_i$ are concurrent or parallel. [i]Nikolai Ivanov Beluhov, Bulgaria[/i]

The centers of the faces of the right rectangular prism shown below are joined to create an octahedron, What is the volume of the octahedron? [asy] import three; size(2inch); currentprojection=orthographic(4,2,2); draw((0,0,0)--(0,0,3),dashed); draw((0,0,0)--(0,4,0),dashed); draw((0,0,0)--(5,0,0),dashed); draw((5,4,3)--(5,0,3)--(5,0,0)--(5,4,0)--(0,4,0)--(0,4,3)--(0,0,3)--(5,0,3)); draw((0,4,3)--(5,4,3)--(5,4,0)); label("3",(5,0,3)--(5,0,0),W); label("4",(5,0,0)--(5,4,0),S); label("5",(5,4,0)--(0,4,0),SE); [/asy] $\textbf{(A) } \dfrac{75}{12} \qquad\textbf{(B) } 10 \qquad\textbf{(C) } 12 \qquad\textbf{(D) } 10\sqrt2 \qquad\textbf{(E) } 15 $

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 $XZY$.

Two triangles $ABC$ and $DEF$ have the same incircle. If a circle passes through $A,B,C,D,E$ prove that it also passes through $F$.

Show that any triangular prism of infinite length can be cut by a plane such that the resulting intersection is an equilateral triangle.

In prism $JHOPKINS$, quadrilaterals $JHOP$ and $KINS$ are parallel and congruent bases that are kites, where $JH = JP = KI = KS$ and $OH = OP = NI = NS$; the longer two sides of each kite have length $\tfrac{4 + \sqrt{5}}{2}$, and the shorter two sides of each kite have length $\tfrac{5 + \sqrt{5}}{4}$. Assume that $\overline{JK}$, $\overline{HI}$, $\overline{ON}$, and $\overline{PS}$ are congruent edges of $JHOPKINS$ perpendicular to the planes containing $JHOP$ and $KINS$. Vertex $J$ is part of a regular pentagon $JAZZ'Y$ that can be inscribed in prism $JHOPKINS$ such that $A \in \overline{HI}$, $Z \in \overline{NI}$, $Z' \in \overline{NS}$, $Y \in \overline{PS}$, $AI = YS$, and $ZI = Z'S$. Compute the height of $JHOPKINS$ (that is, the distance between the bases).

Consider the set of points that are inside or within one unit of a rectangular parallelepiped (box) that measures 3 by 4 by 5 units. Given that the volume of this set is $(m + n \pi)/p$, where $m$, $n$, and $p$ are positive integers, and $n$ and $p$ are relatively prime, find $m + n + p$.

Let $A$ be a solid $a\times b\times c$ rectangular brick, where $a,b,c>0$. Let $B$ be the set of all points which are a distance of at most one from some point of $A$. Express the volume of $B$ as a polynomial in $a,b,$ and $c$.

A right regular hexagonal prism has bases $ABCDEF$, $A'B'C'D'E'F'$ and edges $AA'$, $BB'$, $CC'$, $DD'$, $EE'$, $FF'$, each of which is perpendicular to both hexagons. The height of the prism is $5$ and the side length of the hexagons is $6$. The plane $P$ passes through points $A$, $C'$, and $E$. The area of the portion of $P$ contained in the prism can be expressed as $m\sqrt{n}$, where $n$ is not divisible by the square of any prime. Find $m+n$.

Two walls and the ceiling of a room meet at right angles at point $P$. A fly is in the air one meter from one wall, eight meters from the other wall, and $9$ meters from point $P$. How many meters is the fly from the ceiling? $\textbf{(A) }\sqrt{13}\qquad\textbf{(B) }\sqrt{14}\qquad\textbf{(C) }\sqrt{15}\qquad\textbf{(D) }4\qquad\textbf{(E) }\sqrt{17}$