Let $A_1A_2 \dots A_n$ a cyclic $n$-agon with center $O$ and $P$, $Q$ being two isogonal conjugates of it (i.e, $\angle PA_{i+1}A_i = \angle QA_{i+1}A_{i+2}$ for all $i$). Let $P_i$ be the circumcenter of $\triangle PA_iA_{i+1}$ and $Q_i$ the circumcenter of $\triangle QA_iA_{i+1}$ for all $i$. Prove that: $a) ~P_1P_2 \dots P_n$ and $Q_1Q_2 \dots Q_n$ are cyclic, with centers $O_P$ and $O_Q$, respectively. $b)~O, O_P$ and $O_Q$ are collinears. $c)~O_PO_Q \mid \mid PQ.$ Remark: indices are taken modulo $n$.

Let $n=3k$ be a positive integer (with $k\geq 2$). An equilateral triangle is divided in $n^2$ unit equilateral triangles with sides parallel to the initial, forming a grid. We will call "trapezoid" the trapezoid which is formed by three equilateral triangles (one base is equal to one and the other is equal to two). We colour the points of the grid with three colours (red, blue and green) such that each two neighboring points have different colour. Finally, the colour of a "trapezoid" will be the colour of the midpoint of its big base. Find the number of all "trapezoids" in the grid (not necessarily disjoint) and determine the number of red, blue and green "trapezoids".

Suppose that $x$, $y$, $z$ are real numbers satisfying \[x+y+z=12,\qquad\text{and}\qquad x^2+y^2+z^2=54.\] Prove that:[list](a) Each of the numbers $xy$, $yz$, $zx$ is at least $9$, but at most $25$. (b) One of the numbers $x$, $y$, $z$ is at most $3$, and another one is at least $5$.[/list]

Find the number of ordered pairs $(x,y)$, where $x$ and $y$ are both integers between $1$ and $9$, inclusive, such that the product $x\times y$ ends in the digit $5$. [i]Proposed by Andrew Wen[/i]

In a polygon, every external angle is one sixth of its corresponding internal angle. How many sides does the polygon have?

Let $N$ denote the number of ordered 2011-tuples of positive integers $(a_1,a_2,\ldots,a_{2011})$ with $1\le a_1,a_2,\ldots,a_{2011} \le 2011^2$ such that there exists a polynomial $f$ of degree $4019$ satisfying the following three properties: [list] [*] $f(n)$ is an integer for every integer $n$; [*] $2011^2 \mid f(i) - a_i$ for $i=1,2,\ldots,2011$; [*] $2011^2 \mid f(n+2011) - f(n)$ for every integer $n$. [/list] Find the remainder when $N$ is divided by $1000$. [i]Victor Wang[/i]

Prove that for every positive integer $m$ we can find a finite set $S$ of points in the plane, such that given any point $A$ of $S$, there are exactly $m$ points in $S$ at unit distance from $A$.

Let $ABC$ be a triangle with $AB < AC$. Let $D$ and $E$ be on the lines $CA$ and $BA$, respectively, such that $CD = AB$, $BE = AC$, and $A$, $D$ and $E$ lie on the same side of $BC$. Let $I$ be the incenter of triangle $ABC$, and let $H$ be the orthocenter of triangle $BCI$. Show that $D$, $E$, and $H$ are collinear.

$\indent$ For a positive integer $n$, let $S_n$ be the set of bijective functions from $\{1,2,\dots ,n\}$ to itself. For a pair of positive integers $(a,b)$ such that $1 \leq a <b \leq n$, and for a permutation $\sigma \in S_n$, we say the pair $(a,b)$ is [i][u]expanding[/u][/i] for $\sigma$ if $|\sigma (a)- \sigma(b)| \geq |a-b|$ $\indent$ [b](a)[/b] Is it true that for all integers $n > 1$, there exists $\sigma \in S_n$ so that the number of pairs $(a,b)$ that are expanding for permutation $\sigma$ is less than $1000n\sqrt n$ ? $\indent$ [b](b)[/b] Does there exist a positive integer $n>1$ and a permutation $\sigma \in S_n$ so that the number of pairs $(a,b)$ that are expanding for the permutation $\sigma$ is less than $\frac{n\sqrt n}{1000}$?

$$Problem 1 $$The set S = {1,2,3,...,2006} is partitioned into two disjoint subsets A and B such that: (i) 13 ∈ A; (ii) if a ∈ A, b ∈ B, a+b ∈ S, then a+b ∈ B; (iii) if a ∈ A, b ∈ B, ab ∈ S, then ab ∈ A. Determine the number of elements of A

For the positive integer $k$ we denote by the $a_n$ , the $k$ from the left digit in the decimal notation of the number $2^n$ ($a_n = 0$ if in the notation of the number $2^n$ less than the digits). Consider the infinite decimal fraction $a = \overline{0, a_1a_2a_3...}$. Prove that the number $a$ is irrational.

You are given $25$ pieces of cheese of different weights. Is it always possible to cut one of the pieces into two parts and put the $26$ pieces in two packets so that $\bullet$ each packet contains $13$ pieces; $\bullet$ the total weights of the two packets are equal; $\bullet$ the two parts of the piece which has been cut are in different packets? (VL Dolnikov)

Prove that there are infinitely many positive $n$ that for all prime divisors $p$ of $n^2 + 3, \exists 0 \leq k \leq \sqrt{n}$ and $p \mid k^2+3$

Let $x$ and $y$ are real numbers such that $$\begin{cases} x + y = 2 \\ x^3 + y^3 = 3\end{cases} $$ What is $x^2+y^2$?

In a half-plane, bounded by a line \(r\), equilateral triangles \(S_1, S_2, \ldots, S_n\) are placed, each with one side parallel to \(r\), and their opposite vertex is the point of the triangle farthest from \(r\). For each triangle \(S_i\), let \(T_i\) be its medial triangle. Let \(S\) be the region covered by triangles \(S_1, S_2, \ldots, S_n\), and let \(T\) be the region covered by triangles \(T_1, T_2, \ldots, T_n\). Prove that \[\text{area}(S) \leq 4 \cdot \text{area}(T).\]

Show that for any natural number $n$, the number $A = [\frac{n + 3}{4}] + [ \frac{n + 5}{4} ] + [\frac{n}{2} ] +n^2 + 3n + 3$ is a perfect square. ($[x]$ denotes the integer part of the real number x.)

$n$ people (with names $1,2,\dots,n$) are around a table. Some of them are friends. At each step 2 friend can change their place. Find a necessary and sufficient condition for friendship relation between them that with these steps we can always reach to all of posiible permutations.

Let $ABCD$ be a parallelogram with $\angle BAD < 90^{\circ}$. A circle tangent to sides $\overline{DA}$, $\overline{AB}$, and $\overline{BC}$ intersects diagonal $\overline{AC}$ at points $P$ and $Q$ with $AP < AQ$, as shown. Suppose that $AP = 3$, $PQ = 9$, and $QC = 16$. Then the area of $ABCD$ can be expressed in the form $m\sqrt n$, where $m$ and $n$ are positive integers, and $n$ is not divisible by the square of any prime. Find $m+n$. [asy] defaultpen(linewidth(0.6)+fontsize(11)); size(8cm); pair A,B,C,D,P,Q; A=(0,0); label("$A$", A, SW); B=(6,15); label("$B$", B, NW); C=(30,15); label("$C$", C, NE); D=(24,0); label("$D$", D, SE); P=(5.2,2.6); label("$P$", (5.8,2.6), N); Q=(18.3,9.1); label("$Q$", (18.1,9.7), W); draw(A--B--C--D--cycle); draw(C--A); draw(Circle((10.95,7.45), 7.45)); dot(A^^B^^C^^D^^P^^Q); [/asy]

Prove that for any integer $a \ge 2$ and positive integer $n,$ there exist positive integer $k$ such that $a^k+1,a^k+2,\ldots,a^k+n$ are all composite numbers.

A [i]cross-pentomino[/i] is a shape that consists of a unit square and four other unit squares each sharing a different edge with the first square. If a cross-pentomino is inscribed in a circle of radius $R,$ what is $100R^2$? [i]Author: Ray Li[/i]

Can there be drawn on a circle of radius $1$ a number of $1975$ distinct points, so that the distance (measured on the chord) between any two points (from the considered points) is a rational number?

For any positive integer $n$, let $f(n)$ denote the $n$- th positive nonsquare integer, i.e., $f(1) = 2, f(2) = 3, f(3) = 5, f(4) = 6$, etc. Prove that $f(n)=n +\{\sqrt{n}\}$ where $\{x\}$ denotes the integer closest to $x$. (For example, $\{\sqrt{1}\} = 1, \{\sqrt{2}\} = 1, \{\sqrt{3}\} = 2, \{\sqrt{4}\} = 2$.)

In the coordinate plane a rectangle with vertices $ (0, 0),$ $ (m, 0),$ $ (0, n),$ $ (m, n)$ is given where both $ m$ and $ n$ are odd integers. The rectangle is partitioned into triangles in such a way that [i](i)[/i] each triangle in the partition has at least one side (to be called a “good” side) that lies on a line of the form $ x \equal{} j$ or $ y \equal{} k,$ where $ j$ and $ k$ are integers, and the altitude on this side has length 1; [i](ii)[/i] each “bad” side (i.e., a side of any triangle in the partition that is not a “good” one) is a common side of two triangles in the partition. Prove that there exist at least two triangles in the partition each of which has two good sides.

If the parabola $ y \equal{} \minus{}x^2 \plus{} bx \minus{} 8$ has its vertex on the $ x$-axis, then $ b$ must be: $ \textbf{(A)}\ \text{a positive integer}\qquad \\ \textbf{(B)}\ \text{a positive or a negative rational number}\qquad \\ \textbf{(C)}\ \text{a positive rational number}\qquad \\ \textbf{(D)}\ \text{a positive or a negative irrational number}\qquad \\ \textbf{(E)}\ \text{a negative irrational number}$

$P$ is any point inside a triangle $ABC$. The perimeter of the triangle $AB + BC + Ca = 2s$. Prove that $s < AP +BP +CP < 2s$.