A sphere is inscribed into a prism $ABCA'B'C'$ and touches its lateral faces $BCC'B', CAA'C', ABB'A' $ at points $A_o, B_o, C_o$ respectively. It is known that $\angle A_oBB' = \angle B_oCC' =\angle C_oAA'$. a) Find all possible values of these angles. b) Prove that segments $AA_o, BB_o, CC_o$ concur. c) Prove that the projections of the incenter to $A'B', B'C', C'A'$ are the vertices of a regular triangle.

In a convex $n$-gonal prism all sides are equal. For what $n$ is this prism right?

Consider a prism with triangular base. The total area of the three faces containing a particular vertex $A$ is $K$. Show that the maximum possible volume of the prism is $\sqrt{\frac{K^3}{54}}$ and find the height of this largest prism.

There is a lamp in space.(Consider lamp a point) Do there exist finite number of equal sphers in space that the light of the lamp can not go to the infinite?(If a ray crash in a sphere it stops)

A cylindrical tank with radius $ 4$ feet and height $ 9$ feet is lying on its side. The tank is filled with water to a depth of $ 2$ feet. What is the volume of the water, in cubic feet? $ \textbf{(A)}\ 24\pi \minus{} 36 \sqrt {2} \qquad \textbf{(B)}\ 24\pi \minus{} 24 \sqrt {3} \qquad \textbf{(C)}\ 36\pi \minus{} 36 \sqrt {3} \qquad \textbf{(D)}\ 36\pi \minus{} 24 \sqrt {2} \\ \textbf{(E)}\ 48\pi \minus{} 36 \sqrt {3}$

A regular polyhedron is a polyhedron that is convex and all of its faces are regular polygons. We call a regular polhedron a "[i]Choombam[/i]" iff none of its faces are triangles. a) prove that each choombam can be inscribed in a sphere. b) Prove that faces of each choombam are polygons of at most 3 kinds. (i.e. there is a set $\{m,n,q\}$ that each face of a choombam is $n$-gon or $m$-gon or $q$-gon.) c) Prove that there is only one choombam that its faces are pentagon and hexagon. (Soccer ball) [img]http://aycu08.webshots.com/image/5367/2001362702285797426_rs.jpg[/img] d) For $n>3$, a prism that its faces are 2 regular $n$-gons and $n$ squares, is a choombam. Prove that except these choombams there are finitely many choombams.

Each corner of a rectangular prism is cut off. Two (of the eight) cuts are shown. How many edges does the new figure have? [asy] draw((0,0)--(3,0)--(3,3)--(0,3)--cycle); draw((3,0)--(5,2)--(5,5)--(2,5)--(0,3)); draw((3,3)--(5,5)); draw((2,0)--(3,1.8)--(4,1)--cycle,linewidth(1)); draw((2,3)--(4,4)--(3,2)--cycle,linewidth(1));[/asy] $ \text{(A)}\ 24\qquad\text{(B)}\ 30\qquad\text{(C)}\ 36\qquad\text{(D)}\ 42\qquad\text{(E)}\ 48 $ [i]Assume that the planes cutting the prism do not intersect anywhere in or on the prism.[/i]

A regular triangular prism has the altitude $h,$ and the two bases of the prism are equilateral triangles with side length $a.$ Dream-holes are made in the centers of both bases, and the three lateral faces are mirrors. Assume that a ray of light, entering the prism through the dream-hole in the upper base, then being reflected once by any of the three mirrors, quits the prism through the dream-hole in the lower base. Find the angle between the upper base and the light ray at the moment when the light ray entered the prism, and the length of the way of the light ray in the interior of the prism.

Square corners, $5$ units on a side, are removed from a $20$ unit by $30$ unit rectangular sheet of cardboard. The sides are then folded to form an open box. The surface area, in square units, of the interior of the box is [asy] fill((0,0)--(20,0)--(20,5)--(0,5)--cycle,lightgray); fill((20,0)--(20+5*sqrt(2),5*sqrt(2))--(20+5*sqrt(2),5+5*sqrt(2))--(20,5)--cycle,lightgray); draw((0,0)--(20,0)--(20,5)--(0,5)--cycle); draw((0,5)--(5*sqrt(2),5+5*sqrt(2))--(20+5*sqrt(2),5+5*sqrt(2))--(20,5)); draw((20+5*sqrt(2),5+5*sqrt(2))--(20+5*sqrt(2),5*sqrt(2))--(20,0)); draw((5*sqrt(2),5+5*sqrt(2))--(5*sqrt(2),5*sqrt(2))--(5,5),dashed); draw((5*sqrt(2),5*sqrt(2))--(15+5*sqrt(2),5*sqrt(2)),dashed); [/asy] $\text{(A)}\ 300 \qquad \text{(B)}\ 500 \qquad \text{(C)}\ 550 \qquad \text{(D)}\ 600 \qquad \text{(E)}\ 1000$

Three edges of a parallelepiped lie on three intersecting diagonals of the lateral faces of a triangular prism. Find the ratio of the volumes of the parallelepiped and the prism.

Legs $L_1, L_2, L_3, L_4$ of a square table each have length $n$, where $n$ is a positive integer. For how many ordered 4-tuples $(k_1, k_2, k_3, k_4)$ of nonnegative integers can we cut a piece of length $k_i$ from the end of leg $L_i \; (i=1,2,3,4)$ and still have a stable table? (The table is [i]stable[/i] if it can be placed so that all four of the leg ends touch the floor. Note that a cut leg of length 0 is permitted.)

A real number is assigned to each vertex of a triangular prism so that the number on any vertex is the arithmetic mean of the numbers on the three adjacent vertices. Prove that all six numbers are equal.

Let $P$ be a given (non-degenerate) polyhedron. Prove that there is a constant $c(P)>0$ with the following property: If a collection of $n$ balls whose volumes sum to $V$ contains the entire surface of $P,$ then $n>c(P)/V^2.$

The base of an inclined prism is a triangle $ABC$. The perpendicular projection of $B_1$, one of the top vertices, is the midpoint of $BC$. The dihedral angle between the lateral faces through $BC$ and $AB$ is $\alpha$, and the lateral edges of the prism make an angle $\beta$ with the base. If $r_1, r_2, r_3$ are exradii of a perpendicular section of the prism, assuming that in $ABC, \cos^2 A + \cos^2 B + \cos^2 C = 1, \angle A < \angle B < \angle C,$ and $BC = a$, calculate $r_1r_2 + r_1r_3 + r_2r_3.$

Let $ABCA'B'C'$ a right triangular prism with the bases equilateral triangles. A plane $\alpha$ containing point $A$ intersects the rays $BB'$ and $CC'$ at points E and $F$, so that $S_ {ABE} + S_{ACF} = S_{AEF}$. Determine the measure of the angle formed by the plane $(AEF)$ with the plane $(BCC')$.

Consider the right prism with the rhombus with side $a$ and acute angle $60^{\circ}$ as a base. This prism was intersected by some plane intersecting its side edges, such that the cross-section of the prism and the plane is a square. Determine all possible lengths of the side of this square.

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]

A rectangular prism with positive integer side lengths formed by stacking unit cubes is called [i]bipartisan[/i] if the same number of unit cubes can be seen on the surface as those which cannot be seen on the surface. How many non-congruent bipartisan rectangular prisms are there? [i]2018 CCA Math Bonanza Team Round #8[/i]

Given a regular hexagonal prism. Find a polygonal line that, starting from a vertex of the base, runs through all the lateral faces and ends at the vertex of the face top, located on the same edge as the starting vertex, and has a minimum length.

The $100$ vertices of a prism, whose base is a $50$-gon, are labeled with numbers $1, 2, 3, \ldots, 100$ in any order. Prove that there are two vertices, which are connected by an edge of the prism, with labels differing by not more than $48$. Note: In all the triangles the three vertices do not lie on a straight line.

The segments $[AD], [BE]$ and $[CF]$ are the side edges of the right triangle prism. (the equilateral triangle is a base) Find all the points in its base $ABC$, situated on the equal distances from the $(AE), (BF)$ and $(CD)$ lines.

$A$ and $B$ wish to divide a cake into two pieces. Each wants the largest piece he can get. The cake is a triangular prism with the triangular faces horizontal. $A$ chooses a point $P$ on the top face. $B$ then chooses a vertical plane through the point $P$ to divide the cake. $B$ chooses which piece to take. Which point $P$ should $A $ choose in order to secure as large a slice as possible?

Given a right triangular prism $A_1B_1C_1 - ABC$ with $\angle BAC = \frac{\pi}{2}$ and $AB = AC = AA_1$, let $G$, $E$ be the midpoints of $A_1B_1$, $CC_1$ respectively, and $D$, $F$ be variable points lying on segments $AC$, $AB$ (not including endpoints) respectively. If $GD \bot EF$, the range of the length of $DF$ is ${ \textbf{(A)}\ [\frac{1}{\sqrt{5}}, 1)\qquad\textbf{(B)}\ [\frac{1}{5}, 2)\qquad\textbf{(C)}\ [1, \sqrt{2})\qquad\textbf{(D)}} [\frac{1}{\sqrt{2}}, \sqrt{2})\qquad $

Prove the following: a) If $ABCA'B'C'$ is a right prism and $M \in (BC), N \in (CA), P \in (AB)$ such that $A'M, B'N$ and $C'P$ are perpendicular each other and concurrent, then the prism $ABCA'B'C'$ is regular. b) If $ABCA'B'C'$ is a regular prism and $\frac{AA'}{AB}=\frac{\sqrt6}{4}$ , then there are $M \in (BC), N \in (CA), P \in (AB)$ so that the lines $A'M, B'N$ and $C'P$ are perpendicular each other and concurrent.

Which of the following sets could NOT be the lengths of the external diagonals of a right rectangular prism [a "box"]? (An [i]external diagonal[/i] is a diagonal of one of the rectangular faces of the box.) $ \textbf{(A)}\ \{4, 5, 6\} \qquad\textbf{(B)}\ \{4, 5, 7\} \qquad\textbf{(C)}\ \{4, 6, 7\} \qquad\textbf{(D)}\ \{5, 6, 7\} \qquad\textbf{(E)}\ \{5, 7, 8\} $