Let $ P$ be a convex $ n$ polygon each of which sides and diagnoals is colored with one of $ n$ distinct colors. For which $ n$ does: there exists a coloring method such that for any three of $ n$ colors, we can always find one triangle whose vertices is of $ P$' and whose sides is colored by the three colors respectively.

Find the smallest constant $c$ such that for every $4$ points in a unit square there are two a distance $\leq c$ apart.

$2n$ ($n\ge 2$) teams took part in the football league matches and there were $2n-1$ matchweeks. In each matchweek each team played one match. Any two teams met with each other during the matches in exactly one game. Moreover, in each match one team was the host and the second was a guest. Say a team is [i] traveling[/i], if in any two consecutive matchweeks it was once a host and once a guest. Prove that there are at most two traveling teams.

Let $ABC $be a right triangle with $\angle C = 90^o$ and let $D$ be the foot of the altitude from $C$. Let $E$ be the centroid of triangle $ACD$ and let $F$ be the centroid of triangle $BCD$. The point $P$ satisfies $\angle CEP = 90^o$ and $|CP| = |AP|$, while point $Q$ satisfies $\angle CFQ = 90^o$ and $|CQ| = |BQ|$. Prove that $PQ$ passes through the centroid of triangle $ABC$.

On sides $AB$ and $CB$ of $\vartriangle ABC$ there are drawn equal segments, $AD$ and $CE$, respectively, of arbitrary length (but shorter than min($AB,BC$)). Find the locus of midpoints of all possible segments $DE$.

Determine all pairs of positive integers $(m,n)$ for which an $m\times n$ rectangle can be tiled with (possibly rotated) L-shaped trominos.

For a positive integer $n$ we denote by $d(n)$ the number of positive divisors of $n$ and by $s(n)$ the sum of these divisors. For example, $d(2018)$ is equal to $4$ since $2018$ has four divisors $(1, 2, 1009, 2018)$ and $s(2018) = 1 + 2 + 1009 + 2018 = 3030$. Determine all positive integers $x$ such that $s(x) \cdot d(x) = 96$. (Richard Henner)

[i]Introductory part [/i] We call an $n$-tuple $x = (x_1, x_2, ... , x_n)$, with $x_k \in R$ (or respectively with all $x_k \in Z$) a real vector (or respectively an integer vector). The set of all real vectors (respectively all integer vectors) is usually denoted by $R^n$ (respectively $Z^n$). A vector $x$ is null if and only if $x_k = 0$, for all $k \in \{1, 2,... , n\}$. Also let $U_n$ be the set of all real vectors $x = (x_1, x_2, ... , x_n)$, such that $x^2_1 + x^2_2 + ...+ x^2_n = 1$. For two vectors $x = (x_1, ... , x_n), y = (y_1, ..., y_n)$ we define the scalar product as the real number $x\cdot y = x_1y_1 + x_2y_2 +...+ x_ny_n$. We define the norm of the vector $x$ as $||x|| =\sqrt{x^2_1 + x^2_2 + ...+ x^2_n}$ [i]The problem[/i] Let $A(k, r) = \{x \in U_n |$ for all $z \in Z^n$ we have either $|x \cdot z| \ge \frac{k}{||z||^r}$ or $z$ is null $\}$. Prove that if $r > n - 1$ the we can find a positive number $k$ such that $A(k, r)$ is not empty, and if $r < n - 1$ we cannot find such a positive number $k$.

A line $\ell$ with slope of $\frac{1}{3}$ insects the ellipse $C:\frac{x^2}{36}+\frac{y^2}{4}=1$ at points $A,B$ and the point $P\left( 3\sqrt{2} , \sqrt{2}\right)$ is above the line $\ell$. [list] [b](1)[/b] Prove that the locus of the incenter of triangle $PAB$ is a segment, [b](2)[/b] If $\angle APB=\frac{\pi}{3}$, then find the area of triangle $PAB$.[/list]

Peter writes a positive integer on the whiteboard. Each minute Basil multiplies the last written number by 2 or by 3 and writes the product on the whiteboard too. Can Peter choose the starting integer such that, irrespective of Basil's strategy, at any given moment the number of integers on the whiteboard starting with 1 or 2 would exceed the number of the ones starting with 7, 8 or 9 ? Maxim Didin

Let $k\ge 2$, $n\ge 1$, $a_1, a_2,\dots, a_k$ and $b_1, b_2, \dots, b_n$ be integers such that $1<a_1<a_2<\dots <a_k<b_1<b_2<\dots <b_n$. Prove that if $a_1+a_2+\dots +a_k>b_1+b_2+\dots + b_n$, then $a_1\cdot a_2\cdot \ldots \cdot a_k>b_1\cdot b_2 \cdot \ldots \cdot b_n$.

For a positive integer $K$, define a sequence, $\{a_n\}_n$, as following $a_1=K$, \[ a_{n+1} = \{ \begin{array} {cc} a_n-1 , & \mbox{ if } a_n \mbox{ is even} \\ \frac{a_n-1}2 , & \mbox{ if } a_n \mbox{ is odd} \end{array}, \] for all $n\geq 1$. Find the smallest value of $K$, which makes $a_{2005}$ the first term equal to 0.

The diagonals $AD, BE$ and $CF$ of a convex hexagon concur at a point $M$. Suppose the six triangles $ABM, BCM, CDM, DEM, EFM$ and $FAM$ are all acute-angled and the circumcentre of all these triangles lie on a circle. Prove that the quadrilaterals $ABDE, BCEF$ and $CDFA$ have equal areas.

Let $ A $ be a nonempty set of real numbers, and let be two functions $ f,g:A\longrightarrow A $ having the following properties: $ \text{(i)} f $ is increasing $ \text{(ii)} f-g $ is nonpositive everywhere $ \text{(iii)} f(A)\subset g(A) $ [b]a)[/b] Prove that $ f=g $ if $ A $ is the set of all nonnegative integers. [b]b)[/b] Is true that $ f=g $ if $ A $ is the set of all integers? [i]Dorel Miheț[/i]

Given an integer $n\ge 2$, prove that \[\lfloor \sqrt n \rfloor + \lfloor \sqrt[3]n\rfloor + \cdots +\lfloor \sqrt[n]n\rfloor = \lfloor \log_2n\rfloor + \lfloor \log_3n\rfloor + \cdots +\lfloor \log_nn\rfloor\]. [hide="Edit"] Thanks to shivangjindal for pointing out the mistake (and sorry for the late edit)[/hide]

If $0 < p$, $0 < q$ and $p +q < 1$ prove $$(px + qy)^2 \le px^2 + qy^2$$