$\indent$ 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 $s < AP +BP +CP < 2s$.

For the positive real numbers $ a,b,c$ prove the inequality \[ \frac {c}{a \plus{} 2b} \plus{} \frac {a}{b \plus{} 2c} \plus{} \frac {b}{c \plus{} 2a}\ge1. \]

Let $m$ circles intersect in points $A$ and $B$. We write numbers using the following algorithm: we write $1$ in points $A$ and $B$, in every midpoint of the open arc $AB$ we write $2$, then between every two numbers written in the midpoint we write their sum and so on repeating $n$ times. Let $r(n,m)$ be the number of appearances of the number $n$ writing all of them on our $m$ circles. a) Determine $r(n,m)$; b) For $n=2006$, find the smallest $m$ for which $r(n,m)$ is a perfect square. Example for half arc: $1-1$; $1-2-1$; $1-3-2-3-1$; $1-4-3-5-2-5-3-4-1$; $1-5-4-7-3-8-5-7-2-7-5-8-3-7-4-5-1$...

Find, with proof, all positive integers $n$ for which $2^n + 12^n + 2011^n$ is a perfect square.

For a real number $a$, denote by $(a]$ the smallest integer that is not less than $a$. Find all real values of $x$ for which holds the equality $$(\sin x]^2 + (\cos x]^2 =|tg x| +|ctg x|.$$

The parallelepiped contains a sphere of radius $r$ and is contained within a sphere of radius $R$. Prove that $ \frac{R}{r} \geq \sqrt{3} $.

Prove that the line joining the centroid and the incenter of a non-isosceles triangle is perpendicular to the base if and only if the sum of the other two sides is thrice the base.

Consider a triangle $XYZ$ and a point $O$ in its interior. Three lines through $O$ are drawn, parallel to the respective sides of the triangle. The intersections with the sides of the triangle determine six line segments from $O$ to the sides of the triangle. The lengths of these segments are integer numbers $a, b, c, d, e$ and $f$ (see figure). Prove that the product $a \cdot b \cdot c\cdot d \cdot e \cdot f$ is a perfect square. [asy] unitsize(1 cm); pair A, B, C, D, E, F, O, X, Y, Z; X = (1,4); Y = (0,0); Z = (5,1.5); O = (1.8,2.2); A = extension(O, O + Z - X, X, Y); B = extension(O, O + Y - Z, X, Y); C = extension(O, O + X - Y, Y, Z); D = extension(O, O + Z - X, Y, Z); E = extension(O, O + Y - Z, Z, X); F = extension(O, O + X - Y, Z, X); draw(X--Y--Z--cycle); draw(A--D); draw(B--E); draw(C--F); dot("$A$", A, NW); dot("$B$", B, NW); dot("$C$", C, SE); dot("$D$", D, SE); dot("$E$", E, NE); dot("$F$", F, NE); dot("$O$", O, S); dot("$X$", X, N); dot("$Y$", Y, SW); dot("$Z$", Z, dir(0)); label("$a$", (A + O)/2, SW); label("$b$", (B + O)/2, SE); label("$c$", (C + O)/2, SE); label("$d$", (D + O)/2, SW); label("$e$", (E + O)/2, SE); label("$f$", (F + O)/2, NW); [/asy]

Let $B$ and $C$ be two fixed points in the plane. For each point $A$ of the plane, outside of the line $BC$, let $G$ be the barycenter of the triangle $ABC$. Determine the locus of points $A$ such that $\angle BAC + \angle BGC = 180^{\circ}$. Note: The locus is the set of all points of the plane that satisfies the property.