Let $ABCDEFGH$ be a parallelepiped with $AE \parallel BF \parallel CG \parallel DH$. Prove the inequality \[AF + AH + AC \leq AB + AD + AE + AG.\] In what cases does equality hold? [i]Proposed by France.[/i]

In a right circular cone of wood, the radius of the circumference $T$ of the base circle measures $10$ cm, while every point on said circumference is $20$ cm away. from the apex of the cone. A red ant and a termite are located at antipodal points of $T$. A black ant is located at the midpoint of the segment that joins the vertex with the position of the termite. If the red ant moves to the black ant's position by the shortest possible path, how far does it travel?

Assume that a face of a convex polyhedron $ P$ has a common edge with every other face. Show that there exists a simple closed polygon that consists of edges of $ P$ and passes through all vertices. [i]L .Lovasz[/i]

A right circular cylinder is inscribed in a sphere of radius $\sqrt3$. What is the height of the cylinder when its volume is maximal?

Suppose you have a sphere tangent to the $xy$-plane with its center having positive $z$-coordinate. If it is projected from a point $P=(0,b,a)$ to the $xy$-plane, it gives the conic section $y=x^2$. If we write $a=\tfrac pq$ where $p,q$ are integers, find $p+q$.

Find all positive integer bases $b \ge 9$ so that the number \[ \frac{{\overbrace{11 \cdots 1}^{n-1 \ 1's}0\overbrace{77 \cdots 7}^{n-1\ 7's}8\overbrace{11 \cdots 1}^{n \ 1's}}_b}{3} \] is a perfect cube in base 10 for all sufficiently large positive integers $n$. [i]Proposed by Yang Liu[/i]

The sum of the base-$ 10$ logarithms of the divisors of $ 10^n$ is $ 792$. What is $ n$? $ \textbf{(A)}\ 11\qquad \textbf{(B)}\ 12\qquad \textbf{(C)}\ 13\qquad \textbf{(D)}\ 14\qquad \textbf{(E)}\ 15$

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}$

What is the largest possible radius of a circle contained in a 4-dimensional hypercube of side length 1?

We consider a prism which has the upper and inferior basis the pentagons: $A_{1}A_{2}A_{3}A_{4}A_{5}$ and $B_{1}B_{2}B_{3}B_{4}B_{5}$. Each of the sides of the two pentagons and the segments $A_{i}B_{j}$ with $i,j=1,\ldots,5$ is colored in red or blue. In every triangle which has all sides colored there exists one red side and one blue side. Prove that all the 10 sides of the two basis are colored in the same color.

A cube with an edge of $ n $ cm is divided in $ n^3 $ mini-cubes with edges of legth $ 1 $ cm. Only the exterior of the cube is colored. [b]a)[/b] How many of the mini-cubes haven't any colored face? [b]b)[/b] How many of the mini-cubes have only one colored face? [b]c)[/b] How many of the mini-cubes have, at least, two colored faces? [b]d)[/b] If we draw with blue all the diagonals of all the faces of the cube, upon how many mini-cubes do we find blue segments?

Prove the following statement: There does not exist a pyramid with square base and congruent lateral faces for which the measures of all edges, total area, and volume are integers.

A polyhedron has $12$ faces and is such that: [b][i](i)[/i][/b] all faces are isosceles triangles, [b][i](ii)[/i][/b] all edges have length either $x$ or $y$, [b][i](iii)[/i][/b] at each vertex either $3$ or $6$ edges meet, and [b][i](iv)[/i][/b] all dihedral angles are equal. Find the ratio $x/y.$

Is it possible that the heights of a tetrahedron (that is, a triangular pyramid) would be equal to the numbers $1$, $2$, $3$ and $6$?

While Theophrastus was talking to Aristotle about the classification of plants, had a dog tied to a perfectly smooth cylindrical column of radius $r$, with a very fine rope that wrapped around the column and with a loop. The dog had the extreme free from the rope around his neck. In trying to reach Theophrastus, he put the rope tight and it broke. Find out how far from the column the knot was in the time to break the rope. [hide=original wording]Mientras Teofrasto hablaba con Arist´oteles sobre la clasificaci´on de las plantas, ten´ıa un perro atado a una columna cil´ındrica perfectamente lisa de radio r, con una cuerda muy fina que envolv´ıa la columna y con un lazo. El perro ten´ıa el extremo libre de la cuerda cogido a su cuello. Al intentar alcanzar a Teofrasto, puso la cuerda tirante y ´esta se rompi´o. Averiguar a qu´e distancia de la columna estaba el nudo en el momento de romperse la cuerda.[/hide]

Prove that for any tetrahedron in space, it is possible to find two perpendicular planes such that ratio between the projections of the tetrahedron on the two planes lies in the interval $[\frac{1}{\sqrt{2}}, \sqrt{2}].$

Let $EFGH$, $EFDC$, and $EHBC$ be three adjacent square faces of a cube, for which $EC=8$, and let $A$ be the eighth vertex of the cube. Let $I$, $J$, and $K$, be the points on $\overline{EF}$, $\overline{EH}$, and $\overline{EC}$, respectively, so that $EI=EJ=EK=2$. A solid $S$ is obtained by drilling a tunnel through the cube. The sides of the tunnel are planes parallel to $\overline{AE}$, and containing the edges, $\overline{IJ}$, $\overline{JK}$, and $\overline{KI}$. The surface area of $S$, including the walls of the tunnel, is $m+n\sqrt{p}$, where $m$, $n$, and $p$ are positive integers and $p$ is not divisible by the square of any prime. Find $m+n+p$.

Find the orders of the subgroups of the group of rotations of the cube, and find its normal subgroups.

Let be a tetahedron $ OABC $ with $ OA\perp OB\perp OC\perp OA. $ Show that $$ OH\le r\left( 1+\sqrt 3 \right) , $$ where $ H $ is the orthocenter of $ ABC $ and $ r $ is radius of the inscribed spere of $ OABC. $ [i]Valentin Vornicu[/i]

Consider a cube $ABCDA'B'C'D'$ (where $ABCD$ is a square and $AA' \parallel BB' \parallel CC' \parallel DD'$) and a point $P$ on the line $AA'$. Construct center $S$ of a sphere which has plane $ABB'$ as a plane of symmetry, $P$ lies on the sphere and $p = AB$, $q = A'D'$ are its tangent lines. Discuss conditions of solvability with respect to different position of the point $P$ (on line $AA'$).

Three pairwise orthogonal chords of a sphere $S$ are drawn through a given point $P$ inside $S$. Prove that the sum of the squares of their lengths does not depend on their directions.

Let $S$ be a fixed point on a sphere $\Sigma$ with center $\Omega$. Consider all tetrahedra $SABC$ inscribed in $\Sigma$ such that $SA,SB,SC$ are pairwise orthogonal. (a) Prove that all the planes $ABC$ pass through a single point. (b) In one such tetrahedron, $H$ and $O$ are the orthogonal projections of $S$ and $\Omega$ onto the plane $ABC$, respectively. Let $R$ denote the circumradius of $\triangle ABC$. Prove that $R^2=OH^2+2SH^2$.

Let $ABCD$ be a tetrahedron inscribed in a sphere with center $O$, and $G$ and $I$ be its barycenter and incenter respectively. Prove that the following are equivalent: (i) Points $O$ and $G$ coincide. (ii) The four faces of the tetrahedron are congruent. (iii) Points $O$ and $I$ coincide.

Consider a plane $\epsilon$ and three non-collinear points $A,B,C$ on the same side of $\epsilon$; suppose the plane determined by these three points is not parallel to $\epsilon$. In plane $\epsilon$ take three arbitrary points $A',B',C'$. Let $L,M,N$ be the midpoints of segments $AA', BB', CC'$; Let $G$ be the centroid of the triangle $LMN$. (We will not consider positions of the points $A', B', C'$ such that the points $L,M,N$ do not form a triangle.) What is the locus of point $G$ as $A', B', C'$ range independently over the plane $\epsilon$?

In regular tetrahedron $ABCD$, $E\in AB,F\in CD$, satisfying: $\frac{|AE|}{|EB|}=\frac{|CF|}{|FD|}=\lambda(\lambda\in R_+)$. Note that $f(\lambda)=\alpha_{\lambda}+\beta_{\lambda}$, where $\alpha_{\lambda}=<EF,AC>,\alpha_{\lambda}=<EF,BD>$. $\text{(A)}$ $f(\lambda)$ increases in $(0,+\infty)$ $\text{(B)}$ $f(\lambda)$ decreases in $(0,+\infty)$ $\text{(C)}$ $f(\lambda)$ increases in $(0,1)$, decreases in $(1,+\infty)$ $\text{(D)}$ $f(\lambda)$ is a fixed value in $(0,+\infty)$