Let $ n$ be a positive integer. Given an integer coefficient polynomial $ f(x)$, define its [i]signature modulo $ n$[/i] to be the (ordered) sequence $ f(1), \ldots , f(n)$ modulo $ n$. Of the $ n^n$ such $ n$-term sequences of integers modulo $ n$, how many are the signature of some polynomial $ f(x)$ if a) $ n$ is a positive integer not divisible by the square of a prime. b) $ n$ is a positive integer not divisible by the cube of a prime.

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.

Consider the cube $ABCDA'B'C'D'$. The bisectors of the angles $\angle A' C'A$ and $\angle A' AC'$ intersect $AA'$ and $A'C$ in the points $P$, respectively $S$. The point $M$ is the foot of the perpendicular from $A'$ on $CP$ , and $N$ is the foot of the perpendicular from $A'$ to $AS$. Point $O$ is the center of the face $ABB'A'$ a) Prove that the planes $(MNO)$ and $(AC'B)$ are parallel. b) Calculate the distance between these planes, knowing that $AB = 1$.

We consider a prism with 6 faces, 5 of which are circumscriptible quadrilaterals. Prove that all the faces of the prism are circumscriptible quadrilaterals.

A truncated trigonal pyramid is circumscribed around a sphere touching its bases at points $T_1, T_2$. Let $h$ be the altitude of the pyramid, $R_1, R_2$ be the circumradii of its bases, and $O_1, O_2$ be the circumcenters of the bases. Prove that $$R_1R_2h^2 = (R_1^2-O_1T_1^2)(R_2^2-O_2T_2^2).$$

Consider a prism, not necessarily right, whose base is a rhombus $ABCD$ with side $AB = 5$ and diagonal $AC = 8$. A sphere of radius $r$ is tangent to the plane $ABCD$ at $C$ and tangent to the edges $AA_1$ , $BB _1$ and $DD_ 1$ of the prism. Calculate $r$ .

Let $ABCD$ be a tetrahedron and suppose that $M$ is a point inside it such that $\angle MAD=\angle MBC$ and $\angle MDB=\angle MCA$. Prove that $$MA\cdot MB+MC\cdot MD<\max(AD\cdot BC,AC\cdot BD).$$

Let $r$ be a real such that $0<r\leq 1$. Denote by $V(r)$ the volume of the solid formed by all points of $(x,\ y,\ z)$ satisfying \[x^2+y^2+z^2\leq 1,\ x^2+y^2\leq r^2\] in $xyz$-space. (1) Find $V(r)$. (2) Find $\lim_{r\rightarrow 1-0} \frac{V(1)-V(r)}{(1-r)^{\frac 32}}.$ (3) Find $\lim_{r\rightarrow +0} \frac{V(r)}{r^2}.$

Prove that the four segments connecting the vertices of the tetrahedron with the centers of gravity of the opposite faces have a common point.

Show that any triangular prism of infinite length can be cut by a plane such that the resulting intersection is an equilateral triangle.

A pyramid with a quadrangular base is given such that each pair of circles inscribed in adjacent faces has a common point. Prove that the touchpoints of these circles with the base of the pyramid lie on one circle.

Suppose that every two opposite edges of a tetrahedron are orthogonal. Show that the midpoints of the six edges lie on a sphere.

Consider a circular cone with vertex $ V$, and let $ ABC$ be a triangle inscribed in the base of the cone, such that $ AB$ is a diameter and $ AC \equal{} BC$. Let $ L$ be a point on $ BV$ such that the volume of the cone is 4 times the volume of the tetrahedron $ ABCL$. Find the value of $ BL/LV$.

a) Prove that center of smallest sphere containing a finite subset of $\mathbb R^{n}$ is inside convex hull of the point that lie on sphere. b) $A$ is a finite subset of $\mathbb R^{n}$, and distance of every two points of $A$ is not larger than 1. Find radius of the largest sphere containing $A$.

Alex marked one point on each of the six interior faces of a hollow unit cube. Then he connected by strings any two marked points on adjacent faces. Prove that the total length of these strings is at least $6\sqrt2$.

A solid figure consisting of unit cubes is shown in the picture. Is it possible to exactly fill a cube with these figures if the side length of the cube is a) 15; b) 30?

A cuboctahedron is the convex hull of (smallest convex set containing) the $12$ points $(\pm 1, \pm 1, 0), (\pm 1, 0, \pm 1), (0, \pm 1, \pm 1)$. Find the cosine of the solid angle of one of the triangular faces, as viewed from the origin. (Take a figure and consider the set of points on the unit sphere centered on the origin such that the ray from the origin through the point intersects the figure. The area of that set is the solid angle of the figure as viewed from the origin.)

Determine if there exists a convex polyhedron such that (1) it has 12 edges, 6 faces and 8 vertices; (2) it has 4 faces with each pair of them sharing a common edge of the polyhedron.

The heights of a triangular pyramid intersect at one point. Prove that all flat angles at any vertex of the surface are either acute, or right, or obtuse.

Prove that any rigid flat triangle $T$ of area less than $4$ can be inserted through a triangular hole $Q$ with area $3$.

Let $x_1, x_2, x_3, x_4$ be the distances from an arbitrary point inside the tetrahedron to the planes of its faces, and let $h_1, h_2, h_3, h_4$ be the corresponding heights of the tetrahedron. Prove that $$\sqrt{h_1+h_2+h_3+h_4} \ge \sqrt{x_1}+\sqrt{x_2}+\sqrt{x_3}+\sqrt{x_4}$$ (Dmitry Nomirovsky)

Let $ ABC$ be an equilateral triangle with side length equal to $ N \in \mathbb{N}.$ Consider the set $ S$ of all points $ M$ inside the triangle $ ABC$ satisfying \[ \overrightarrow{AM} \equal{} \frac{1}{N} \cdot \left(n \cdot \overrightarrow{AB} \plus{} m \cdot \overrightarrow{AC} \right)\] with $ m, n$ integers, $ 0 \leq n \leq N,$ $ 0 \leq m \leq N$ and $ n \plus{} m \leq N.$ Every point of S is colored in one of the three colors blue, white, red such that [b](i) [/b]no point of $ S \cap [AB]$ is coloured blue [b](ii)[/b] no point of $ S \cap [AC]$ is coloured white [b](iii)[/b] no point of $ S \cap [BC]$ is coloured red Prove that there exists an equilateral triangle the following properties: [b](1)[/b] the three vertices of the triangle are points of $ S$ and coloured blue, white and red, respectively. [b](2)[/b] the length of the sides of the triangle is equal to 1. [i]Variant:[/i] Same problem but with a regular tetrahedron and four different colors used.

Lines that are drawn perpendicular to the faces of a triangular pyramid through the centers of the inscribed circles intersect at one point. Prove that the sums of the opposite edges of such a pyramid are equal to each other.

A sphere is inscribed in a cube that has a surface area of $ 24$ square meters. A second cube is then inscribed within the sphere. What is the surface area in square meters of the inner cube? $ \textbf{(A)}\ 3 \qquad \textbf{(B)}\ 6 \qquad \textbf{(C)}\ 8 \qquad \textbf{(D)}\ 9 \qquad \textbf{(E)}\ 12$

Prove that the four lines, joining the vertices of a tetrahedron with the incenters of the opposite faces, have a common point if and only if the three products of the lengths of opposite sides are equal.