A parallelepiped has surface area 216 and volume 216. Show that it is a cube.

A point $P$ in the interior of a convex polyhedron in Euclidean space is called a [i]pivot point[/i] of the polyhedron if every line through $P$ contains exactly $0$ or $2$ vertices of the polyhedron. Determine, with proof, the maximum number of pivot points that a polyhedron can contain.

$480$ $1$ cm unit cubes are used to build a block measuring $6$ cm by $8$ cm by $10$ cm. A tiny ant then chews his way in a straight line from one vertex of the block to the furthest vertex. How many cubes does the ant pass through? The ant is so tiny that he does not "pass through" cubes if he is merely passing through where their edges or vertices meet. [i]2017 CCA Math Bonanza Individual Round #11[/i]

Three diagonals of a regular $n$-gon prism intersect at an interior point $O$. Show that $O$ is the center of the prism. (The diagonal of the prism is a segment joining two vertices not lying on the same face of the prism.)

Let $V$ be a rectangular prism with integer side lengths. The largest face has area $240$ and the smallest face has area $48$. A third face has area $x$, where $x$ is not equal to $48$ or $240$. What is the sum of all possible values of $x$?

The solid shown has a square base of side length $s$. The upper edge is parallel to the base and has length $2s$. All other edges have length $s$. Given that $s = 6 \sqrt{2}$, what is the volume of the solid? [asy] import three; size(170); pathpen = black+linewidth(0.65); pointpen = black; currentprojection = perspective(30,-20,10); real s = 6 * 2^.5; triple A=(0,0,0),B=(s,0,0),C=(s,s,0),D=(0,s,0),E=(-s/2,s/2,6),F=(3*s/2,s/2,6); draw(F--B--C--F--E--A--B); draw(A--D--E, dashed); draw(D--C, dashed); label("$2s$", (s/2, s/2, 6), N); label("$s$", (s/2, 0, 0), SW); [/asy]

Let $\mathsf{S}_1,\mathsf{S}_2$ be concentric spheres with radii $a,b$ respectively, where $a<b.$ Denote $ABCDA'B'C'D'$ a square cuboid ($ABCD,A'B'C'D$ are the squares and $AA'\parallel BB'\parallel CC'\parallel DD'$) such that $A,B,C,D\in\mathsf{S}_2$ and the plane $A'B'C'D'$ is tangent to $\mathsf{S}_1.$ Finally assume that \[\frac{AB}{AA'}=\frac ab.\] Compute the lengths $AB,AA'.$ How many of such cuboids exist (up to a congruence)?

Consider the regular quadrilateral prism $ABCDA'B'C 'D'$, in which $AB = a,AA' = \frac{a \sqrt {2}}{2}$, and $M$ is the midpoint of $B' C'$. Let $F$ be the foot of the perpendicular from $B$ on line $MC$, Let determine the measure of the angle between the planes $(BDF)$ and $(HBS)$.

[asy] import olympiad; import cse5; size(5cm); pointpen = black; pair A = Drawing((10,17.32)); pair B = Drawing((0,0)); pair C = Drawing((20,0)); draw(A--B--C--cycle); pair X = 0.85*A + 0.15*B; pair Y = 0.82*A + 0.18*C; pair W = (-11,0) + X; pair Z = (19, 9); draw(W--X, EndArrow); draw(X--Y, EndArrow); draw(Y--Z, EndArrow); anglepen=black; anglefontpen=black; MarkAngle("\theta", C,Y,Z, 3); [/asy] The cross-section of a prism with index of refraction $1.5$ is an equilateral triangle, as shown above. A ray of light comes in horizontally from air into the prism, and has the opportunity to leave the prism, at an angle $\theta$ with respect to the surface of the triangle. Find $\theta$ in degrees and round to the nearest whole number. [i](Ahaan Rungta, 5 points)[/i]

The right prism $[A_1A_2A_3\ldots A_nA_1'A_2'A_3'\ldots A_n'],n\in\mathbb{N},n\ge 3$, has a convex polygon as its base. It is known that $A_1A_2'\perp A_2A_3',A_2A_3'\perp A_3A_4',$$\ldots A_{n-1}A_n'\perp A_nA_1', A_nA_1'\perp A_1A_2'$. Show that: $a)$ $n=3$; $b)$ the prism is regular.

Isabella is making sushi. She slices a piece of salmon into the shape of a solid triangular prism. The prism is $2$ cm thick, and its triangular faces have side lengths $7$ cm, $ 24$cm, and $25$ cm. Find the volume of this piece of salmon in cm$^3$. [i]Proposed by Isabella Li[/i]

Distinct planes $p_1,p_2,....,p_k$ intersect the interior of a cube $Q$. Let $S$ be the union of the faces of $Q$ and let $ P =\bigcup_{j=1}^{k}p_{j} $. The intersection of $P$ and $S$ consists of the union of all segments joining the midpoints of every pair of edges belonging to the same face of $Q$. What is the difference between the maximum and minimum possible values of $k$? $ \textbf{(A)}\ 8\qquad\textbf{(B)}\ 12\qquad\textbf{(C)}\ 20\qquad\textbf{(D)}\ 23\qquad\textbf{(E)}\ 24 $

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.

For which natural number $n$ is it possible to place natural number from 1 to $3n$ on the edges of a right $n$-angled prism (on each edge there is exactly one number placed and each one is used exactly 1 time) in such a way, that the sum of all the numbers, that surround each face is the same?

The base of a prism is an $n$-gon. We wish to colour its $2n$ vertices in three colours in such a way that every vertex is connected by edges to vertices of all three colours. (a) Prove that if $n$ is divisible by $3$, then the task is possible. {b) Prove that if the task is possible, then $n$ is divisible by $3$. (A Shapovalov)

Points $E$ and $F$ are taken on the diagonals $AB_1$ and $CA_1$ of the lateral faces $ABB_1A_1$ and $CAA_1C_1$ of a triangular prism $ABCA_1B_1C_1$ so that $EF\parallel BC_1$. Find the ratio of the lengths of $EF$ and $BC_1$.

A hollow box (with negligible thickness) shaped like a rectangular prism has a volume of $108$ cubic units. The top of the box is removed, exposing the faces on the inside of the box. What is the minimum possible value for the sum of the areas of the faces on the outside and inside of the box?

An open box is constructed by starting with a rectangular sheet of metal $ 10$ in. by $ 14$ in. and cutting a square of side $ x$ inches from each corner. The resulting projections are folded up and the seams welded. The volume of the resulting box is: $ \textbf{(A)}\ 140x \minus{} 48x^2 \plus{} 4x^3 \qquad \textbf{(B)}\ 140x \plus{} 48x^2 \plus{} 4x^3\qquad \\\textbf{(C)}\ 140x \plus{} 24x^2 \plus{} x^3\qquad \textbf{(D)}\ 140x \minus{} 24x^2 \plus{} x^3\qquad \textbf{(E)}\ \text{none of these}$

Find the minimum possible sum of lengths of edges of a prism all of whose edges are tangent of a unit sphere. [Muller-Pfeiffer].

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 bases of a right hexagonal prism are regular hexagons of side length $s > 0$, and the prism has height $h$. The prism contains some water, and when it is placed on a flat surface with a hexagonal face on the bottom, the water has depth $\frac{s\sqrt3}{4}$. The water depth doesn’t change when the prism is turned so that a rectangular face is on the bottom. Compute $\frac{h}{s}$.

Let $\mathcal{B}$ be the set of rectangular boxes that have volume $23$ and surface area $54$. Suppose $r$ is the least possible radius of a sphere that can fit any element of $\mathcal{B}$ inside it. Then $r^{2}$ can be expressed as $\tfrac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.

A right rectangular prism has integer side lengths $a$, $b$, and $c$. If $\text{lcm}(a,b)=72$, $\text{lcm}(a,c)=24$, and $\text{lcm}(b,c)=18$, what is the sum of the minimum and maximum possible volumes of the prism? [i]Proposed by Deyuan Li and Andrew Milas[/i]

Let the $ABCA'B'C'$ be a regular prism. The points $M$ and $N$ are the midpoints of the sides $BB'$, respectively $BC$, and the angle between the lines $AB'$ and $BC'$ is of $60^\circ$. Let $O$ and $P$ be the intersection of the lines $A'C$ and $AC'$, with respectively $B'C$ and $C'N$. a) Prove that $AC' \perp (OPM)$; b) Find the measure of the angle between the line $AP$ and the plane $(OPM)$. [i]Mircea Fianu[/i]

We say that a prism is [i]binary[/i] if there exists a labelling of the vertices of the prism with integers from the set $\{-1,1\}$ such that the product of the numbers assigned to the vertices of each face (base or lateral face) is equal to $-1$. a) Prove that any [i]binary[/i] prism has the number of total vertices divisible by 8; b) Prove that any prism with 2000 vertices is [i]binary[/i].