Find the largest possible integer $k$, such that the following statement is true: Let $2009$ arbitrary non-degenerated triangles be given. In every triangle the three sides are coloured, such that one is blue, one is red and one is white. Now, for every colour separately, let us sort the lengths of the sides. We obtain \[ \left. \begin{array}{rcl} & b_1 \leq b_2\leq\ldots\leq b_{2009} & \textrm{the lengths of the blue sides }\\ & r_1 \leq r_2\leq\ldots\leq r_{2009} & \textrm{the lengths of the red sides }\\ \textrm{and } & w_1 \leq w_2\leq\ldots\leq w_{2009} & \textrm{the lengths of the white sides }\\ \end{array}\right.\] Then there exist $k$ indices $j$ such that we can form a non-degenerated triangle with side lengths $b_j$, $r_j$, $w_j$. [i]Proposed by Michal Rolinek, Czech Republic[/i]

We are given the triangle $ABC$ with an area of $2000$. Let $P,Q,R$ be the midpoints of the sides $BC$, $AC$, $AB$. Let $U,V,W$ be the midpoints of the sides $QR$, $PR$, $PQ$. The lengths of the line segments $AU$, $BV$, $CW$ are $x$, $y$, $z$. Show that there exists a triangle with side lengths $x$, $y$ and $z$ and caluclate it's area.

A soldier needs to check if there are any mines in the interior or on the sides of an equilateral triangle $ABC.$ His detector can detect a mine at a maximum distance equal to half the height of the triangle. The soldier leaves from one of the vertices of the triangle. Which is the minimum distance that he needs to traverse so that at the end of it he is sure that he completed successfully his mission?

$C$, $C'$ are concentric circles with radii $R$, $R'$. A rectangle has two adjacent vertices on $C$ and the other two vertices on $C'$. Find its sides if its area is as large as possible.

The side lengths $a,$ $b,$ $c$ of a triangle $ABC$ form an arithmetical progression (such that $b-a=c-b$). The side lengths $a_{1},$ $b_{1},$ $c_{1}$ of a triangle $A_{1}B_{1}C_{1}$ also form an arithmetical progression (with $b_{1}-a_{1}=c_{1}-b_{1}$). [Hereby, $a=BC,$ $b=CA,$ $c=AB, $ $a_{1}=B_{1}C_{1},$ $b_{1}=C_{1}A_{1},$ $c_{1}=A_{1}B_{1}.$] Moreover, we know that $\measuredangle CAB=\measuredangle C_{1}A_{1}B_{1}.$ Show that triangles $ABC$ and $A_{1}B_{1}C_{1}$ are similar.

Show that if $a, b, c$ are the lengths of the sides of a triangle and if $2S = a + b + c$, then \[\frac{a^n}{b+c} + \frac{b^n}{c+a} +\frac{c^n}{a+b} \geq \left(\dfrac 23 \right)^{n-2}S^{n-1} \quad \forall n \in \mathbb N \] [i]Proposed by Greece.[/i]

Determine all positive integers $n$ with the following property: for each triple $(a, b, c)$ of positive real numbers there is a triple $(k, \ell, m)$ of non-negative integer numbers so that $an^k$, $bn^{\ell}$ and $cn^m$ are the lengths of the sides of a (non-degenerate) triangle shapes.

(a) Let $ABC$ be any triangle. Let $X, Y, Z$ be points on the sides $BC, CA, AB$ respectively. Suppose that $BX \leq XC, CY \leq YA, AZ \leq ZB$. Show that the area of the triangle $XYZ$ $\geq 1\slash 4$ times the area of $ABC.$ (b) Let $ABC$ be any triangle, and let $X, Y, Z$ be points on the sides $BC, CA, AB$ respectively. Using (a) or by any other method, show: One of the three corner triangles $AZY, BXZ, CYX$ has an area $\leq$ area of the triangle $XYZ.$

Let $a_1,\ldots,a_n$ and $b_1\ldots,b_n$ be $2n$ real numbers. Prove that there exists an integer $k$ with $1\le k\le n$ such that $ \sum_{i=1}^n|a_i-a_k| ~~\le~~ \sum_{i=1}^n|b_i-a_k|.$ (Proposed by Gerhard Woeginger, Austria)

In a unit square one hundred segments are drawn from the centre to the sides dividing the square into one hundred parts (triangles and possibly quadruilaterals). If all parts have equal perimetr $p$, show that $\dfrac{14}{10} < p < \dfrac{15}{10}$.

Segments $AA'$, $BB'$, and $CC'$ are the bisectrices of triangle $ABC$. It is known that these lines are also the bisectrices of triangle $A'B'C'$. Is it true that triangle $ABC$ is regular?

Consider $ n$ points in a plane which are vertices of a convex polygon. Prove that the set of the lengths of the sides and the diagonals of the polygon has at least $ \lfloor n/2\rfloor$ elements.

A trapezoid with bases $AD$ and $BC$ is circumscribed about a circle, $E$ is the intersection point of the diagonals. Prove that $\angle AED$ is not acute.

Let $ABC$ be a triangle, and let $r$ denote its inradius. Let $R_A$ denote the radius of the circle internally tangent at $A$ to the circle $ABC$ and tangent to the line $BC$; the radii $R_B$ and $R_C$ are defined similarly. Show that $\frac{1}{R_A} + \frac{1}{R_B} + \frac{1}{R_C}\leq\frac{2}{r}$.

Let be given two concentric circles of radii $R$ and $R_1 > R$. Let quadrilateral $ABCD$ is inscribed in the smaller circle and let the rays $CD, DA, AB, BC$ meet the larger circle at $A_1, B_1, C_1, D_1$ respectively. Prove that $$ \frac{\sigma(A_1B_1C_1D_1)}{\sigma(ABCD)} \geq \frac{R_1^2}{R^2}$$ where $\sigma(P)$ denotes the area of a polygon $P.$

What is the largest possible distance between two points, one on the sphere of radius 19 with center $(-2, -10, 5)$ and the other on the sphere of radius 87 with center $(12, 8, -16)$?

Let $ABC$ be a triangle and $M,N,P$ be the midpoints of sides $BC, CA, AB$. The lines $AM, BN, CP$ meet the circumcircle of $ABC$ in the points $A_{1}, B_{1}, C_{1}$. Show that the area of triangle $ABC$ is at most the sum of areas of triangles $BCA_{1}, CAB_{1}, ABC_{1}$.

Establish the maximum and minimum values that the sum $|a| + |b| + |c|$ can have if $a, b, c$ are real numbers such that the maximum value of $|ax^2 + bx + c|$ is $1$ for $-1 \leq x \leq 1.$

Prove that the sum of the lengths of the medians of a triangle is at least three quarters of its perimeter.

A soldier needs to check if there are any mines in the interior or on the sides of an equilateral triangle $ABC.$ His detector can detect a mine at a maximum distance equal to half the height of the triangle. The soldier leaves from one of the vertices of the triangle. Which is the minimum distance that he needs to traverse so that at the end of it he is sure that he completed successfully his mission?

Let $T_1$ be a triangle having $a, b, c$ as lengths of its sides and let $T_2$ be another triangle having $u, v,w$ as lengths of its sides. If $P,Q$ are the areas of the two triangles, prove that \[16PQ \leq a^2(-u^2 + v^2 + w^2) + b^2(u^2 - v^2 + w^2) + c^2(u^2 + v^2 - w^2).\] When does equality hold?

Two equal-sized regular $n$-gons intersect to form a $2n$-gon $C$. Prove that the sum of the sides of $C$ which form part of one $n$-gon equals half the perimeter of $C$. [i]Alternative formulation:[/i] Let two equal regular $n$-gons $S$ and $T$ be located in the plane such that their intersection $S\cap T$ is a $2n$-gon (with $n\ge 3$). The sides of the polygon $S$ are coloured in red and the sides of $T$ in blue. Prove that the sum of the lengths of the blue sides of the polygon $S\cap T$ is equal to the sum of the lengths of its red sides.

Given a triangle $ \,ABC,\,$ let $ \,I\,$ be the center of its inscribed circle. The internal bisectors of the angles $ \,A,B,C\,$ meet the opposite sides in $ \,A^{\prime },B^{\prime },C^{\prime }\,$ respectively. Prove that \[ \frac {1}{4} < \frac {AI\cdot BI\cdot CI}{AA^{\prime }\cdot BB^{\prime }\cdot CC^{\prime }} \leq \frac {8}{27}. \]

Let $D$ be the closed unit disk in the plane, and let $z_1,z_2,\ldots,z_n$ be fixed points in $D$. Prove that there exists a point $z$ in $D$ such that the sum of the distances from $z$ to each of the $n$ points is greater or equal than $n$.

Given a circle and a point $A$ inside the circle, but not at its center. Find points $B$, $C$, $D$ on the circle which maximise the area of the quadrilateral $ABCD$.