Club Truncator is in a soccer league with six other teams, each of which it plays once. In any of its 6 matches, the probabilities that Club Truncator will win, lose, or tie are each $\frac{1}{3}$. The probability that Club Truncator will finish the season with more wins than losses is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Find all prime numbers $p$ such that $16p + 1$ is a perfect cube.

A $4\times 4\times h$ rectangular box contains a sphere of radius $2$ and eight smaller spheres of radius $1$. The smaller spheres are each tangent to three sides of the box, and the larger sphere is tangent to each of the smaller spheres. What is $h$? [asy] import graph3; import solids; real h=2+2*sqrt(7); currentprojection=orthographic((0.75,-5,h/2+1),target=(2,2,h/2)); currentlight=light(4,-4,4); draw((0,0,0)--(4,0,0)--(4,4,0)--(0,4,0)--(0,0,0)^^(4,0,0)--(4,0,h)--(4,4,h)--(0,4,h)--(0,4,0)); draw(shift((1,3,1))*unitsphere,gray(0.85)); draw(shift((3,3,1))*unitsphere,gray(0.85)); draw(shift((3,1,1))*unitsphere,gray(0.85)); draw(shift((1,1,1))*unitsphere,gray(0.85)); draw(shift((2,2,h/2))*scale(2,2,2)*unitsphere,gray(0.85)); draw(shift((1,3,h-1))*unitsphere,gray(0.85)); draw(shift((3,3,h-1))*unitsphere,gray(0.85)); draw(shift((3,1,h-1))*unitsphere,gray(0.85)); draw(shift((1,1,h-1))*unitsphere,gray(0.85)); draw((0,0,0)--(0,0,h)--(4,0,h)^^(0,0,h)--(0,4,h)); [/asy] $\textbf{(A) }2+2\sqrt 7\qquad \textbf{(B) }3+2\sqrt 5\qquad \textbf{(C) }4+2\sqrt 7\qquad \textbf{(D) }4\sqrt 5\qquad \textbf{(E) }4\sqrt 7\qquad$

Finitely many points are given on the surface of a sphere, such that every four of them lie on the surface of open hemisphere. Prove that all points lie on the surface of an open hemisphere.

The cube $ABCDA' B' C' D' $has of length a. Consider the points $K \in [AB], L \in [CC' ], M \in [D'A']$. a) Show that $\sqrt3 KL \ge KB + BC + CL$ b) Show that the perimeter of triangle $KLM$ is strictly greater than $2a\sqrt3$.

Each face of a cube is painted either red or blue, each with probability $ 1/2$. The color of each face is determined independently. What is the probability that the painted cube can be placed on a horizontal surface so that the four vertical faces are all the same color? $ \textbf{(A)}\ \frac14 \qquad \textbf{(B)}\ \frac{5}{16} \qquad \textbf{(C)}\ \frac38 \qquad \textbf{(D)}\ \frac{7}{16} \qquad \textbf{(E)}\ \frac12$

Let $a,b,c,d$ be the areas of the triangular faces of a tetrahedron, and let $h_a, h_b, h_c, h_d$ be the corresponding altitudes of the tetrahedron. If $V$ denotes the volume of tetrahedron, prove that \[ (a+b+c+d)(h_a+h_b+h_c+h_d) \geq 48V \]

Two planes, $ \alpha $ and $ \beta, $ form a dihedral angle of $ 30^{\circ} , $ and their intersection is the line $ d. $ A point $ A $ situated at the exterior of this angle projects itself in $ P\not\in d $ on $ \alpha , $ and in $ Q\not\in d $ on $ \beta $ such that $ AQ<AP. $ Name $ B $ the projection of $ A $ upon $ d. $ [b]a)[/b] Are $ A,B,P,Q, $ coplanar? [b]b)[/b] Knowing that a perpendicular to $ \beta $ make with $ AB $ an angle of $ 60^{\circ} , $ and $ AB=4, $ find the area of $ BPQ. $

How many ways are there to fill an $8\times8\times8$ cube with $1\times1\times8$ sticks? Rotations and reflections are considered distinct. [i]Team #2[/i]

We consider a fixed point $P$ in the interior of a fixed sphere$.$ We construct three segments $PA, PB,PC$, perpendicular two by two$,$ with the vertexes $A, B, C$ on the sphere$.$ We consider the vertex $Q$ which is opposite to $P$ in the parallelepiped (with right angles) with $PA, PB, PC$ as edges$.$ Find the locus of the point $Q$ when $A, B, C$ take all the positions compatible with our problem.

A regular $n$-gonal truncated pyramid is circumscribed around a sphere. Denote the areas of the base and the lateral surfaces of the pyramid by $S_1, S_2$, and $S$, respectively. Let $\sigma$ be the area of the polygon whose vertices are the tangential points of the sphere and the lateral faces of the pyramid. Prove that \[\sigma S = 4S_1S_2 \cos^2 \frac{\pi}{n}.\]

Prove that a pyramid $A_1A_2 \ldots A_{2k+1}S$ with equal lateral edges and equal space angles between adjacent lateral walls is regular.

Let $ABCD$ be a tetrahedron. (a) Prove that the midpoints of the edges $AB,AC,BD$, and $CD$ lie in a plane. (b) Find the point in that plane, whose sum of distances from the lines $AD$ and $BC$ is minimal.

Call a convex polyhedron a [i]footballoid [/i] if it has the following properties. (1) Any face is either a regular pentagon or a regular hexagon. (2) All neighbours of a pentagonal face are hexagonal (a [i]neighbour [/i] of a face is a face that has a common edge with it). Find all possibilities for the number of pentagonal and hexagonal faces of a footballoid.

Let $ T$ be a triangulation of an $ n$-dimensional sphere, and to each vertex of $ T$ let us assign a nonzero vector of a linear space $ V$. Show that if $ T$ has an $ n$-dimensional simplex such that the vectors assigned to the vertices of this simplex are linearly independent, then another such simplex must also exist. [i]L. Lovasz[/i]