Given are $50$ points in the plane, no three of them belonging to a same line. Each of these points is colored using one of four given colors. Prove that there is a color and at least $130$ scalene triangles with vertices of that color.

Consider a function $f:\mathbb N\rightarrow \mathbb N$ such that for any two positive integers $x,y$, the equation $f(xf(y))=yf(x)$ holds. Find the smallest possible value of $f(2007)$.

$S_{1},S_{2},S_{3}$ are three spheres in $\mathbb R^{3}$ that their centers are not collinear. $k\leq8$ is the number of planes that touch three spheres. $A_{i},B_{i},C_{i}$ is the point that $i$-th plane touch the spheres $S_{1},S_{2},S_{3}$. Let $O_{i}$ be circumcenter of $A_{i}B_{i}C_{i}$. Prove that $O_{i}$ are collinear.

The numbers $1$ and $2$ are written on an empty blackboard. Whenever the numbers $m$ and $n$ appear on the blackboard the number $m + n + mn$ may be written. Can we obtain : (1) $13121$, (2) $12131$?

In a certain game, the set $\{1, 2, \dots, 10\}$ is partitioned into equally-sized sets $A$ and $B$. In each of five consecutive rounds, Alice and Bob simultaneously choose an element from $A$ or $B$, respectively, that they have not yet chosen; whoever chooses the larger number receives a point, and whoever obtains three points wins the game. Determine the probability that Alice is guaranteed to win immediately after the set is initially partitioned.

Given a convex polygon with $n$ sides and perimeter $S$, which has an incircle $\omega$ with radius $R$. A regular polygon with $n$ sides, whose vertices lie on $\omega$, has a perimeter $s$. Determine whether the following inequality holds: \[ S \ge \frac{2sRn}{\sqrt{4n^2R^2-s^2}}. \]

The triangle $ ABC$ is given. On the extension of the side $ AB$ we construct the point $ R$ with $ BR \equal{} BC$, where $ AR > BR$ and on the extension of the side $ AC$ we construct the point $ S$ with $ CS \equal{} CB$, where $ AS > CS$. Let $ A_1$ be the point of intersection of the diagonals of the quadrilateral $ BRSC$. Analogous we construct the point $ T$ on the extension of the side $ BC$, where $ CT \equal{} CA$ and $ BT > CT$ and on the extension of the side $ BA$ we construct the point $ U$ with $ AU \equal{} AC$, where $ BU > AU$. Let $ B_1$ be the point of intersection of the diagonals of the quadrilateral $ CTUA$. Likewise we construct the point $ V$ on the extension of the side $ CA$, where $ AV \equal{} AB$ and $ CV > AV$ and on the extension of the side $ CB$ we construct the point $ W$ with $ BW \equal{} BA$ and $ CW > BW$. Let $ C_1$ be the point of intersection of the diagonals of the quadrilateral $ AVWB$. Show that the area of the hexagon $ AC_1BA_1CB_1$ is equal to the sum of the areas of the triangles $ ABC$ and $ A_1B_1C_1$.

Ziquan makes a drawing in the plane for art class. He starts by placing his pen at the origin, and draws a series of line segments, such that the $n^{th}$ line segment has length $n$. He is not allowed to lift his pen, so that the end of the $n^{th}$ segment is the start of the $(n + 1)^{th}$ segment. Line segments drawn are allowed to intersect and even overlap previously drawn segments. After drawing a finite number of line segments, Ziquan stops and hands in his drawing to his art teacher. He passes the course if the drawing he hands in is an $N$ by $N$ square, for some positive integer $N$, and he fails the course otherwise. Is it possible for Ziquan to pass the course?

Let $BE$ and $CF$ be the bisectors of $\angle B$ and $\angle C$ of a triangle $ABC$ whose incentre is $I$. Suppose $EF$, extended, meets the circumcircle of $ABC$ in $M,N$. Show that the circumradius of $MIN$ is twice that of $ABC$.

Find all nonnegative integer numbers such that $7^x- 2 \cdot 5^y = -1$

In an acute-angled triangle $ABC$, $M$ is the midpoint of side $BC$, and $D, E$ and $F$ the feet of the altitudes from $A, B$ and $C$, respectively. Let $H$ be the orthocenter of $\Delta ABC$, $S$ the midpoint of $AH$, and $G$ the intersection of $FE$ and $AH$. If $N$ is the intersection of the median $AM$ and the circumcircle of $\Delta BCH$, prove that $\angle HMA = \angle GNS$. [i]Proposed by Marko Djikic[/i]

Determine the smallest positive integer $a$ for which there exist a prime number $p$ and a positive integer $b \ge 2$ such that $$\frac{a^p -a}{p}=b^2.$$

Consider the following statement: for any permutation $\pi_1\not=\mathbb{I}$ of $\{1,2,...,n\}$ there is a permutation $\pi_2$ such that any permutation on these numbers can be obtained by a finite compostion of $\pi_1$ and $\pi_2$. (a) Prove the statement for $n=3$ and $n=5$. (b) Disprove the statement for $n=4$.

For $i = 1, 2$ let $\vartriangle A_iB_iC_i$ be a triangle with side lengths $a_i, b_i, c_i$ and altitude lengths $p_i, q_i, r_i$. Define $a_3 =\sqrt{a_1^2 + a_2^2}, b_3 =\sqrt{b_1^2 + b_2^2}$ , and $c_3 =\sqrt{c_1^2 + c_2^2}$. Prove that $a_3, b_3, c_3$ are side lengths of a triangle, and if $p_3, q_3, r_3$ are the lengths of altitudes of this triangle, then $p_3^2 \ge p_1^2 +p_2^2$, $q_3^2 \ge q_1^2 +q_2^2$ , and $r_3^2 \ge r_1^2 +r_2^2$

[b]7.[/b] Find the finite groups having only one proper maximal subgroup. [b](A.12)[/b]

Let $a,b,c$ be integers which are lengths of sides of a triangle, $\gcd(a,b,c)=1$ and all the values $$\frac{a^2+b^2-c^2}{a+b-c},\quad\frac{b^2+c^2-a^2}{b+c-a},\quad\frac{c^2+a^2-b^2}{c+a-b}$$ are integers as well. Show that $(a+b-c)(b+c-a)(c+a-b)$ or $2(a+b-c)(b+c-a)(c+a-b)$ is a perfect square.

Let $(a_n)_{n\ge 1}$ and $(b_n)_{n\ge 1}$ be two sequences of natural numbers such that $a_{n+1} = na_n + 1, b_{n+1} = nb_n - 1$ for every $n\ge 1$. Show that these two sequences can have only a finite number of terms in common.

Point $P$ lies on the side $CD$ of the cyclic quadrilateral $ABCD$ such that $\angle CBP = 90^{\circ}$. Let $K$ be the intersection of $AC,BP$ such that $AK = AP = AD$. $H$ is the projection of $B$ onto the line $AC$. Prove that $\angle APH = 90^{\circ}$. [i]Proposed by Iman Maghsoudi - Iran[/i]

Let $a > 3$ be an odd integer. Show that for every positive integer $n$ the number $a^{2^n}- 1$ has at least $n + 1$ distinct prime divisors.

Let $n$ be a positive integer, and let $a_1,a_2,..,a_n$ be pairwise distinct positive integers. Show that $$\sum_{k=1}^{n}{\frac{1}{[a_1,a_2,…,a_k]}} <4,$$ where $[a_1,a_2,…,a_k]$ is the least common multiple of the integers $a_1,a_2,…,a_k$.

When the polynomial $x^3-2$ is divided by the polynomial $x^2-2$, the remainder is $\textbf{(A)} \ 2 \qquad \textbf{(B)} \ -2 \qquad \textbf{(C)} \ -2x-2 \qquad \textbf{(D)} \ 2x+2 \qquad \textbf{(E)} \ 2x-2$

In a $5 \times 5$ checkered square, the middle row and middle column are colored gray. You leave the corner cell and move to the cell adjacent to the side with each move. For each transition from a gray cell to a gray one you need to pay a ruble. What is the smallest number of rubles you need to pay to go around all the squares of the board exactly once (it is not necessary to return to the starting square)?

Prove that every partition of $3$-dimensional space into three disjoint subsets has the following property: One of these subsets contains all possible distances; i.e., for every $a \in \mathbb R^+$, there are points $M$ and $N$ inside that subset such that distance between $M$ and $N$ is exactly $a.$

Find all pairs of integers $(m,n)$ such that $m^6 = n^{n+1} + n -1$.

Let $ ABC$ be a triangle with $ \left|AB\right| \equal{} 14$, $ \left|BC\right| \equal{} 12$, $ \left|AC\right| \equal{} 10$. Let $ D$ be a point on $ \left[AC\right]$ and $ E$ be a point on $ \left[BC\right]$ such that $ \left|AD\right| \equal{} 4$ and $ Area\left(ABC\right) \equal{} 2Area\left(CDE\right)$. Find $ Area\left(ABE\right)$. $\textbf{(A)}\ 4\sqrt {6} \qquad\textbf{(B)}\ 6\sqrt {2} \qquad\textbf{(C)}\ 3\sqrt {6} \qquad\textbf{(D)}\ 4\sqrt {2} \qquad\textbf{(E)}\ 4\sqrt {5}$