Pete has $n^3$ white cubes of the size $1\times1\times1$. He wants to construct a $n\times n\times n$ cube with all its faces being completely white. Find the minimal number of the faces of small cubes that Basil must paint (in black colour) in order to prevent Pete from fulfilling his task. Consider the cases: a) $n = 3$; [i](3 points)[/i] b) $n = 1000$. [i](3 points)[/i]

Determine the planar finite configurations $C$ consisting of at least $3$ points, satisfying the following conditions; if $x$ and $y$ are distinct points of $C$, there exist $z\in C$ such that $xyz$ are three vertices of equilateral triangles

Let $I$ and $O$ be the incenter and the circumcenter of a triangle $ABC$, respectively, and let $s_a$ be the exterior bisector of angle $\angle BAC$. The line through $I$ perpendicular to $IO$ meets the lines $BC$ and $s_a$ at points $P$ and $Q$, respectively. Prove that $IQ = 2IP$.

You are given some equilateral triangles and squares, all with side length 1, and asked to form convex $n$ sided polygons using these pieces. If both types must be used, what are the possible values of $n$, assuming that there is sufficient supply of the pieces?

Find all reals $z$ such that $z^4 - z^3 - 2z^2 - 3z - 1= 0$.

In the plane we consider rectangles whose sides are parallel to the coordinate axes and have positive length. Such a rectangle will be called a [i]box[/i]. Two boxes [i]intersect[/i] if they have a common point in their interior or on their boundary. Find the largest $ n$ for which there exist $ n$ boxes $ B_1$, $ \ldots$, $ B_n$ such that $ B_i$ and $ B_j$ intersect if and only if $ i\not\equiv j\pm 1\pmod n$. [i]Proposed by Gerhard Woeginger, Netherlands[/i]

In the tetrahedron $ABCD$, the point $E$ is the projection of the point $D$ on the plane $(ABC)$. Prove that the following statements are equivalent: a) $C = E$ or $CE \parallel AB$ b) For each point M belonging to the segment $CD$, the following equation is satisfied $$S^2_{\vartriangle ABM}= \frac{CM^2}{CD^2}\cdot S^2_{\vartriangle ABD}+\left(1- \frac{CM^2}{CD^2} \right)S^2_{\vartriangle ABC}$$ where $S_{\vartriangle XYZ}$ means the area of ​​triangle $XYZ$.

A hexagon $ABCDEF$ is inscribed in a circle. Let $P, Q, R, S$ be intersections of $AB$ and $DC$, $BC$ and $ED$, $CD$ and $FE$, $DE$ and $AF$, then $\angle BPC=50^{\circ}$, $\angle CQD=45^{\circ}$, $\angle DRE=40^{\circ}$, $\angle ESF=35^{\circ}$. Let $T$ be an intersection of $BE$ and $CF$. Find $\angle BTC$.

Let $k$ be a positive integer. Show that exists positive integer $n$, such that sets $A = \{ 1^2, 2^2, 3^2, ...\}$ and $B = \{1^2 + n, 2^2 + n, 3^2 + n, ... \}$ have exactly $k$ common elements.

A positive integer $ n$ is given. In an association consisting of $ n$ members work $ 6$ commissions. Each commission contains at least $ \large \frac{n}{4}$ persons. Prove that there exist two commissions containing at least $ \large \frac{n}{30}$ persons in common.

Let $f(x) = ax^2+bx+c$ be a quadratic trinomial with $a$,$b$,$c$ reals such that any quadratic trinomial obtained by a permutation of $f$'s coefficients has an integer root (including $f$ itself). Show that $f(1)=0$.

Find the number of triples of nonnegative integers $(a,b,c)$ satisfying $a + b + c = 300$ and \[ a^2b + a^2c + b^2a + b^2c + c^2a + c^2b = 6{,}000{,}000.\]

Show that all complex roots of the polynomial $P(z)=a_0z^n+a_1z^{n-1}+\ldots+a_{n-1}z+a_n$, where $0<a_0<\ldots<a_n$, satisfy $|z|>1$.

From the vertices of a regular 27-gon, seven are chosen arbitrarily. Prove that among these seven points there are three points that form an isosceles triangle or four points that form an isosceles trapezoid.

[b](1)[/b] Find the number of positive integer solutions $(m,n,r)$ of the indeterminate equation $mn+nr+mr=2(m+n+r)$. [b](2)[/b] Given an integer $k (k>1)$, prove that indeterminate equation $mn+nr+mr=k(m+n+r)$ has at least $3k+1$ positive integer solutions $(m,n,r)$.

A square is dissected into $n$ congruent non-convex polygons whose sides are parallel to the sides of the square, and no two of these polygons are parallel translates of each other. What is the maximum value of $n$? (4)

What are the possible areas of a hexagon with all angles equal and sides $1, 2, 3, 4, 5$, and $6$, in some order?

Alan is standing in the middle of a very long straight road. In addition, there is typically British fog. And he lost his bomb somewhere on the street. He doesn't know how far they are from him away or in which direction it is, and could not see it until it would be no more than $10$ meters away from him. Since he wants to be efficient, he only wants to search at most ten times the distance that the bomb was initially away from him. How he was able to to accomplish this? [hide=original wording]Alan steht mitten auf einer sehr langen geraden Straße. Zudem herrscht typisch britischer Nebel und er hat seine Bombe irgendwo auf der Straße verloren. Er weiß nicht, wie weit sie von ihm entfernt ist oder in welcher Richtung sie liegt, und k¨onnte sie auch erst sehen, wenn sie h¨ochstens 10 Meter von ihm entfernt w¨are. Da er effizient sein will, m¨ochte er maximal eine zehnmal so hohe Distanz auf der Suche zuru¨cklegen, wie die Bombe anfangs von ihm entfernt war. Wie k¨onnte er dies bewerkstelligen¿[/hide]

$n$ is a natural number. for every positive real numbers $x_{1},x_{2},...,x_{n+1}$ such that $x_{1}x_{2}...x_{n+1}=1$ prove that: $\sqrt[x_{1}]{n}+...+\sqrt[x_{n+1}]{n} \geq n^{\sqrt[n]{x_{1}}}+...+n^{\sqrt[n]{x_{n+1}}}$

Let $n$ be an integer greater than $1$.Find all pairs of integers $(s,t)$ such that equations: $x^n+sx=2007$ and $x^n+tx=2008$ have at least one common real root.

For what positive integers $n, k$ (with $k < n$) are the binomial coefficients $${n \choose k- 1} \,\,\, , \,\,\, {n \choose k} \,\,\, , \,\,\, {n \choose k + 1}$$ three successive terms of an arithmetic progression?

The number $10!$ $(10$ is written in base $10)$, when written in the base $12$ system, ends in exactly $k$ zeroes. The value of $k$ is $\textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }3\qquad\textbf{(D) }4\qquad \textbf{(E) } 5$

Let $\triangle ABC$ have $AB=6$, $BC=7$, and $CA=8$, and denote by $\omega$ its circumcircle. Let $N$ be a point on $\omega$ such that $AN$ is a diameter of $\omega$. Furthermore, let the tangent to $\omega$ at $A$ intersect $BC$ at $T$, and let the second intersection point of $NT$ with $\omega$ be $X$. The length of $\overline{AX}$ can be written in the form $\tfrac m{\sqrt n}$ for positive integers $m$ and $n$, where $n$ is not divisible by the square of any prime. Find $100m+n$. [i]Proposed by David Altizio[/i]

In right-angle triangle $ABC$, $\angle C=90$°, Point $D$ is the midpoint of side $AB$. Points $M$ and $C$ lie on the same side of $AB$ such that $MB\bot AB$, line $MD$ intersects side $AC$ at $N$, line $MC$ intersects side $AB$ at $E$. Show that $\angle DBN=\angle BCE$.

Let $Z,W,\lambda$ be complex numbers, $|\lambda|\neq1$. Which statements are correct about the equation $\overline{Z}-\lambda Z=W$? I. $Z=\frac{\overline{\lambda}W+\overline{W}}{1-|\lambda|^2}$ is a solution to the equation. II. The equation has only one solution. III. The equation has two solutions. IV. The equation has infinitely many solutions. $\text{(A)}$ Only I and II. $\text{(B)}$ Only I and III. $\text{(C)}$ Only I and IV. $\text{(D)}$ None of $\text{(A)(B)(C)}$.