The numbers from $ 1$ to $ 8$ are placed at the vertices of a cube in such a manner that the sum of the four numbers on each face is the same. What is this common sum? $ \textbf{(A)}\ 14 \qquad \textbf{(B)}\ 16 \qquad \textbf{(C)}\ 18 \qquad \textbf{(D)}\ 20 \qquad \textbf{(E)}\ 24$

A unit cube has vertices $P_1, P_2, P_3, P_4, P_1', P_2', P_3'$, and $P_4'$. Vertices $P_2, P_3$, and $P_4$ are adjacent to $P_1$, and for $1\leq i\leq 4$, vertices $P_i$ and $P_i'$ are opposite to each other. A regular octahedron has one vertex in each of the segments $P_1P_2, P_1P_3, P_1P_4, P_1'P_2', P_1'P_3'$, and $P_1'P_4'$. What is the octahedron's side length? [asy] import three; size(7.5cm); triple eye = (-4, -8, 3); currentprojection = perspective(eye); triple[] P = {(1, -1, -1), (-1, -1, -1), (-1, 1, -1), (-1, -1, 1), (1, -1, -1)}; // P[0] = P[4] for convenience triple[] Pp = {-P[0], -P[1], -P[2], -P[3], -P[4]}; // draw octahedron triple pt(int k){ return (3*P[k] + P[1])/4; } triple ptp(int k){ return (3*Pp[k] + Pp[1])/4; } draw(pt(2)--pt(3)--pt(4)--cycle, gray(0.6)); draw(ptp(2)--pt(3)--ptp(4)--cycle, gray(0.6)); draw(ptp(2)--pt(4), gray(0.6)); draw(pt(2)--ptp(4), gray(0.6)); draw(pt(4)--ptp(3)--pt(2), gray(0.6) + linetype("4 4")); draw(ptp(4)--ptp(3)--ptp(2), gray(0.6) + linetype("4 4")); // draw cube for(int i = 0; i < 4; ++i){ draw(P[1]--P[i]); draw(Pp[1]--Pp[i]); for(int j = 0; j < 4; ++j){ if(i == 1 || j == 1 || i == j) continue; draw(P[i]--Pp[j]); draw(Pp[i]--P[j]); } dot(P[i]); dot(Pp[i]); dot(pt(i)); dot(ptp(i)); } label("$P_1$", P[1], dir(P[1])); label("$P_2$", P[2], dir(P[2])); label("$P_3$", P[3], dir(-45)); label("$P_4$", P[4], dir(P[4])); label("$P'_1$", Pp[1], dir(Pp[1])); label("$P'_2$", Pp[2], dir(Pp[2])); label("$P'_3$", Pp[3], dir(-100)); label("$P'_4$", Pp[4], dir(Pp[4])); [/asy] $ \textbf{(A)}\ \frac{3\sqrt{2}}{4}\qquad\textbf{(B)}\ \frac{7\sqrt{6}}{16}\qquad\textbf{(C)}\ \frac{\sqrt{5}}{2}\qquad\textbf{(D)}\ \frac{2\sqrt{3}}{3}\qquad\textbf{(E)}\ \frac{\sqrt{6}}{2} $

A solid box is $ 15$ cm by $ 10$ cm by $ 8$ cm. A new solid is formed by removing a cube $ 3$ cm on a side from each corner of this box. What percent of the original volume is removed? $ \textbf{(A)}\ 4.5 \qquad \textbf{(B)}\ 9 \qquad \textbf{(C)}\ 12 \qquad \textbf{(D)}\ 18 \qquad \textbf{(E)}\ 24$

Jesse cuts a circular paper disk of radius $12$ along two radii to form two sectors, the smaller having a central angle of $120$ degrees. He makes two circular cones, using each sector to form the lateral surface of a cone. What is the ratio of the volume of the smaller cone to that of the larger? $ \textbf{(A)}\ \frac{1}{8} \qquad\textbf{(B)}\ \frac{1}{4} \qquad\textbf{(C)}\ \frac{\sqrt{10}}{10} \qquad\textbf{(D)}\ \frac{\sqrt{5}}{6} \qquad\textbf{(E)}\ \frac{\sqrt{10}}{5} $

A cube side 3 is divided into 27 unit cubes. The unit cubes are arbitrarily labeled 1 to 27 (each cube is given a different number). A move consists of swapping the cube labeled 27 with one of its 6 neighbours. Is it possible to find a finite sequence of moves at the end of which cube 27 is in its original position, but cube $n$ has moved to the position originally occupied by $27-n$ (for each $n = 1, 2, \ldots , 26$)?

There are various points in space $ A, B, C_0, C_1, C_2 $, with $ |AC_i| = 2 |BC_i| $ for $ i = 0,1,2 $ and $ |C_1C_2|=\frac{4}{3}|AB| $. Prove that the angle $ C_1C_0C_2 $ is right and the points $ A, B, C_1, C_2 $ lie on one plane.

All edges of the triangular pyramid $ABCD$ are equal in length. Let $M$ be the midpoint of $DB$, $N$ is the point on $\overline{AB}$, such that $2NA=NB$ and $N\not\in AB$ and $P$ is a point on the altitude through point $D$ in $\triangle BCD$. Find $\angle MPD$ if the intersection of the pyramid with the plane $(NMP)$ is a trapezoid.

Let $ f$ be a continuous function on the unit interval $ [0,1]$. Show that \[ \lim_{n \rightarrow \infty} \int_0^1... \int_0^1f(\frac{x_1+...+x_n}{n})dx_1...dx_n=f(\frac12)\] and \[ \lim_{n \rightarrow \infty} \int_0^1... \int_0^1f (\sqrt[n]{x_1...x_n})dx_1...dx_n=f(\frac1e).\]

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.

How many $2 \times 2 \times 2$ cubes must be added to a $8 \times 8 \times 8$ cube to form a $12 \times 12 \times 12$ cube? [i]Proposed by Evan Chen[/i]

Let $ABCDA'B'C'D'$ be a cube (where $ABCD$ and $A'B'C'D'$ are faces and $AA',BB',CC',DD'$ are edges). Consider the four lines $AA', BC, D'C'$ and the line joining the midpoints of $BB'$ and $DD'$. Show that there is no line which cuts all the four lines.

Can we arrange $5$ points in space to form a pentagon with equal sides such that the angle between each pair of adjacent edges is $90^o$?

A rectangular box $\mathcal{P}$ has distinct edge lengths $a, b,$ and $c$. The sum of the lengths of all $12$ edges of $\mathcal{P}$ is $13$, the sum of the areas of all $6$ faces of $\mathcal{P}$ is $\frac{11}{2}$, and the volume of $\mathcal{P}$ is $\frac{1}{2}$. What is the length of the longest interior diagonal connecting two vertices of $\mathcal{P}$? $\textbf{(A)}~2\qquad\textbf{(B)}~\frac{3}{8}\qquad\textbf{(C)}~\frac{9}{8}\qquad\textbf{(D)}~\frac{9}{4}\qquad\textbf{(E)}~\frac{3}{2}$

[5] Find all integers $n$ for which $\frac{n^3+8}{n^2-4}$ is an integer.

The positive integers $ x$ and $ y$ are the two smallest positive integers for which the product of $ 360$ and $ x$ is a square and the product of $ 360$ and $ y$ is a cube. What is the sum of $ x$ and $ y$? $ \textbf{(A)}\ 80 \qquad \textbf{(B)}\ 85 \qquad \textbf{(C)}\ 115 \qquad \textbf{(D)}\ 165 \qquad \textbf{(E)}\ 610$

In space there is a line $ k $ and a cube with a vertex $ M $ and edges $ \overline{MA} $, $ \overline{MB} $, $ \overline{MC} $, of length$ 1$. Prove that the length of the orthogonal projection of edge $ MA $ on the line $ k $ is equal to the area of the orthogonal projection of a square with sides $ MB $ and $ MC $ onto a plane perpendicular to the line $ k $. [hide=original wording]W przestrzeni dana jest prosta $ k $ oraz sześcian o wierzchołku $ M $ i krawędziach $ \overline{MA} $, $ \overline{MB} $, $ \overline{MC} $, długości 1. Udowodnić, że długość rzutu prostokątnego krawędzi $ MA $ na prostą $ k $ jest równa polu rzutu prostokątnego kwadratu o bokach $ MB $ i $ MC $ na płaszczyznę prostopadłą do prostej $ k $.[/hide]

A $10\times10\times10$ cube has $1000$ unit cubes. $500$ of them are coloured black and $500$ of them are coloured white. Show that there are at least $100$ unit squares, being the common face of a white and a black unit cube.

A cubic block with dimensions $n$ by $n$ by $n$ is made up of a collection of $1$ by $1$ by $1$ unit cubes. What is the smallest value of $n$ so that if the outer layer of unit cubes are removed from the block, more than half the original unit cubes will still remain?

Let $ABCD$ be a tetrahedron such that $AD \perp BD$, $BD \perp CD$, $CD \perp AD$ and $AD=10$, $BD=15$, $CD=20$. Let $E$ and $F$ be points such that $E$ lies on $BC$, $DE \perp BC$, and $ADEF$ is a rectangle. If $S$ is the solid consisting of all points inside $ABCD$ but outside $FBCD$, compute the volume of $S$. [i]2015 CCA Math Bonanza Individual Round #13[/i]

Given a tetrahedron $ABCD$ with edges $|AD|=|BC|= a$, $|AC|=|BD|=b$, $AB=c$ and $|CD| = d$. Determine the smallest value of the sum $|AX|+|BX|+|CX|+|DX|$, where $X$ is any point in space.

Denote by $ \mathcal{C}$ the family of all configurations $ C$ of $ N > 1$ distinct points on the sphere $ S^2,$ and by $ \mathcal{H}$ the family of all closed hemispheres $ H$ of $ S^2.$ Compute: $ \displaystyle\max_{H\in\mathcal{H}}\displaystyle\min_{C\in\mathcal{C}}|H\cap C|$, $ \displaystyle\min_{H\in\mathcal{H}}\displaystyle\max_{C\in\mathcal{C}}|H\cap C|$ $ \displaystyle\max_{C\in\mathcal{C}}\displaystyle\min_{H\in\mathcal{H}}|H\cap C|$ and $ \displaystyle\min_{C\in\mathcal{C}}\displaystyle\max_{H\in\mathcal{H}}|H\cap C|.$

Does there exist a cube in space such that the perpendiculars dropped from its eight vertices to a given plane are of length $0, 1, 2, 3, 4, 5, 6$ and $7$? (V Proizvolov)

The numbers $\frac{1}{1}, \frac{1}{2}, \cdots , \frac{1}{2012}$ are written on the blackboard. Aïcha chooses any two numbers from the blackboard, say $x$ and $y$, erases them and she writes instead the number $x + y + xy$. She continues to do this until only one number is left on the board. What are the possible values of the final number?

Find the maximum possible volume of a tetrahedron which has three faces with area $1$.

Among all orthogonal projections of a regular tetrahedron to all possible planes, find the projection of the greatest area.