Let $ x,y,z$ be real numbers. Find the minimum value of $ x^2\plus{}y^2\plus{}z^2$ if $ x^3\plus{}y^3\plus{}z^3\minus{}3xyz\equal{}1$.

Let $O$ be a point of three-dimensional space and let $l_1, l_2, l_3$ be mutually perpendicular straight lines passing through $O$. Let $S$ denote the sphere with center $O$ and radius $R$, and for every point $M$ of $S$, let $S_M$ denote the sphere with center $M$ and radius $R$. We denote by $P_1, P_2, P_3$ the intersection of $S_M$ with the straight lines $l_1, l_2, l_3$, respectively, where we put $P_i \neq O$ if $l_i$ meets $S_M$ at two distinct points and $P_i = O$ otherwise ($i = 1, 2, 3$). What is the set of centers of gravity of the (possibly degenerate) triangles $P_1P_2P_3$ as $M$ runs through the points of $S$?

The insphere and the exsphere opposite to the vertex $D$ of a (not necessarily regular) tetrahedron $ABCD$ touch the face $ABC$ in the points $X$ and $Y$, respectively. Show that $\measuredangle XAB=\measuredangle CAY$.

Four cones are given with a common vertex and the same generatrix, but with, generally speaking, different radii of the bases. Each of them is tangent to two others. Prove that the four tangent points of the circles of the bases of the cones lie on the same circle.

Define a $\emph{big circle}$ in a sphere as a circle that has two diametrically oposite points of the sphere in it. Suppose $(AB)$ as the big circle that passes through $A$ and $B$. Also, let a $\emph{Spheric Triangle}$ be $3$ connected by big circles. The angle between two circles that intersect is defined by the angle between the two tangent lines from the intersection point through the two circles in their respective planes. Define also $\angle XYZ$ the angle between $(XY)$ and $(YZ)$. Two circles are tangent if the angle between them is 0. All the points in the following problem are in a sphere S. Let $\Delta ABC$ be a spheric triangle with all its angles $<90^{\circ}$ such that there is a circle $\omega$ tangent to $(BC)$,$(CA)$,$(AB)$ in $D,E,F$. Show that there is $P\in S$ with $\angle PAB=\angle DAC$, $\angle PCA=\angle FCB$, $\angle PBA=\angle EBC$.

Points $ A$, $ B$, $ C$, $ D$, and $ E$ are located in 3-dimensional space with $ AB \equal{} BC \equal{} CD \equal{} DE \equal{} EA \equal{} 2$ and $ \angle ABC \equal{} \angle CDE \equal{} \angle DEA \equal{} 90^\circ.$ The plane of $ \triangle ABC$ is parallel to $ \overline{DE}$. What is the area of $ \triangle BDE$? $ \textbf{(A)}\ \sqrt2 \qquad \textbf{(B)}\ \sqrt3 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ \sqrt5 \qquad \textbf{(E)}\ \sqrt6$

Supppose five points in a plane are situated so that no two of the straight lines joining them are parallel, perpendicular, or coincident. From each point perpendiculars are drawn to all the lines joining the other four points. Determine the maxium number of intersections that these perpendiculars can have.

A cube has edge length $2$. Suppose that we glue a cube of edge length $1$ on top of the big cube so that one of its faces rests entirely on the top face of the larger cube. The percent increase in the surface area (sides, top, and bottom) from the original cube to the new solid formed is closest to [asy] draw((0,0)--(2,0)--(3,1)--(3,3)--(2,2)--(0,2)--cycle); draw((2,0)--(2,2)); draw((0,2)--(1,3)); draw((1,7/3)--(1,10/3)--(2,10/3)--(2,7/3)--cycle); draw((2,7/3)--(5/2,17/6)--(5/2,23/6)--(3/2,23/6)--(1,10/3)); draw((2,10/3)--(5/2,23/6)); draw((3,3)--(5/2,3)); [/asy] $\text{(A)}\ 10 \qquad \text{(B)}\ 15 \qquad \text{(C)}\ 17 \qquad \text{(D)}\ 21 \qquad \text{(E)}\ 25$

Prove that the orthogonal projections of the vertex $ D $ of the tetrahedron $ ABCD $ onto the bisectors of the internal and external dihedral angles at the edges $ \overline{AB} $, $ \overline{BC} $ and $ \overline{CA} $ belong to one plane .

Let $ABCD$ be a regular tetrahedron with side length $2$. The plane parallel to edges $AB$ and $CD$ and lying halfway between them cuts $ABCD$ into two pieces. Find the surface area of one of these pieces.

Show that the sum of the distances from any point of a regular tetrahedron to its faces is equal to the height of this tetrahedron.

Eight congruent equilateral triangles, each of a different color, are used to construct a regular octahedron. How many distinguishable ways are there to construct the octahedron? (Two colored octahedrons are distinguishable if neither can be rotated to look just like the other.) [asy]import three; import math; size(180); defaultpen(linewidth(.8pt)); currentprojection=orthographic(2,0.2,1); triple A=(0,0,1); triple B=(sqrt(2)/2,sqrt(2)/2,0); triple C=(sqrt(2)/2,-sqrt(2)/2,0); triple D=(-sqrt(2)/2,-sqrt(2)/2,0); triple E=(-sqrt(2)/2,sqrt(2)/2,0); triple F=(0,0,-1); draw(A--B--E--cycle); draw(A--C--D--cycle); draw(F--C--B--cycle); draw(F--D--E--cycle,dotted+linewidth(0.7));[/asy]$ \textbf{(A)}\ 210 \qquad \textbf{(B)}\ 560 \qquad \textbf{(C)}\ 840 \qquad \textbf{(D)}\ 1260 \qquad \textbf{(E)}\ 1680$

Let $VA_1A_2\ldots A_n$ be a pyramid, where $n\ge 4$. A plane $\Pi$ intersects the edges $VA_1,VA_2,\ldots, VA_n$ at the points $B_1,B_2,\ldots,B_n$ respectively such that the polygons $A_1A_2\ldots A_n$ and $B_1B_2\ldots B_n$ are similar. Prove that the plane $\Pi$ is parallel to the plane containing the base $A_1A_2\ldots A_n$. [i]Laurentiu Panaitopol[/i]

What is the largest possible height of a right cylinder with radius $3$ that can fit in a cube with side length $12$?

[b]a)[/b] Show that there are $ 4 $ equidistant parallel planes that passes through the vertices of the same tetrahedron. [b]b)[/b] How many such $ \text{4-tuplets} $ of planes does exist, in function of the tetrahedron?