Let $\mathbb{N}_0$ and $\mathbb{Z}$ be the set of all non-negative integers and the set of all integers, respectively. Let $f:\mathbb{N}_0\rightarrow\mathbb{Z}$ be a function defined as \[f(n)=-f\left(\left\lfloor\frac{n}{3}\right\rfloor \right)-3\left\{\frac{n}{3}\right\} \] where $\lfloor x \rfloor$ is the greatest integer smaller than or equal to $x$ and $\{ x\}=x-\lfloor x \rfloor$. Find the smallest integer $n$ such that $f(n)=2010$.

Consider the sequence $(a_n)_{n\in \mathbb{N}}$ with $a_0=a_1=a_2=a_3=1$ and $a_na_{n-4}=a_{n-1}a_{n-3} + a^2_{n-2}$. Prove that all the terms of this sequence are integer numbers.

You are given a set of $n$ blocks, each weighing at least $1$; their total weight is $2n$. Prove that for every real number $r$ with $0 \leq r \leq 2n-2$ you can choose a subset of the blocks whose total weight is at least $r$ but at most $r + 2$.

Let $n > 1$ be an integer. Find, with proof, all sequences $x_1 , x_2 , \ldots , x_{n-1}$ of positive integers with the following three properties: (a). $x_1 < x_2 < \cdots < x_{n-1}$ ; (b). $x_i + x_{n-i} = 2n$ for all $i = 1, 2, \ldots , n - 1$; (c). given any two indices $i$ and $j$ (not necessarily distinct) for which $x_i + x_j < 2n$, there is an index $k$ such that $x_i + x_j = x_k$.

Given a natural number $n \geq 2$, consider all the fractions of the form $\frac{1}{ab}$, where $a$ and $b$ are natural numbers, relative primes and such that: $a < b \leq n$, $a+b>n$. Show that for each $n$, the sum of all this fractions are $\frac12$.

Figures $ 0$, $ 1$, $ 2$, and $ 3$ consist of $ 1$, $ 5$, $ 13$, and $ 25$ nonoverlapping squares, respectively. If the pattern were continued, how many nonoverlapping squares would there be in figure $ 100$? [asy] unitsize(8); draw((0,0)--(1,0)--(1,1)--(0,1)--cycle); draw((9,0)--(10,0)--(10,3)--(9,3)--cycle); draw((8,1)--(11,1)--(11,2)--(8,2)--cycle); draw((19,0)--(20,0)--(20,5)--(19,5)--cycle); draw((18,1)--(21,1)--(21,4)--(18,4)--cycle); draw((17,2)--(22,2)--(22,3)--(17,3)--cycle); draw((32,0)--(33,0)--(33,7)--(32,7)--cycle); draw((29,3)--(36,3)--(36,4)--(29,4)--cycle); draw((31,1)--(34,1)--(34,6)--(31,6)--cycle); draw((30,2)--(35,2)--(35,5)--(30,5)--cycle); label("Figure",(0.5,-1),S); label("$0$",(0.5,-2.5),S); label("Figure",(9.5,-1),S); label("$1$",(9.5,-2.5),S); label("Figure",(19.5,-1),S); label("$2$",(19.5,-2.5),S); label("Figure",(32.5,-1),S); label("$3$",(32.5,-2.5),S);[/asy]$ \textbf{(A)}\ 10401 \qquad \textbf{(B)}\ 19801 \qquad \textbf{(C)}\ 20201 \qquad \textbf{(D)}\ 39801 \qquad \textbf{(E)}\ 40801$

Let $x_1,x_2,\ldots,x_n$ be real numbers. Prove that \[ \sum_{i,j=1}^n |x_i+x_j|\geq n\sum_{i=1}^n |x_i| \]

For all integers $n\geq 1$ we define $x_{n+1}=x_1^2+x_2^2+\cdots +x_n^2$, where $x_1$ is a positive integer. Find the least $x_1$ such that 2006 divides $x_{2006}$.

Two teams, $ A$ and $ B$, fight for a territory limited by a circumference. $ A$ has $ n$ blue flags and $ B$ has $ n$ white flags ($ n\geq 2$, fixed). They play alternatively and $ A$ begins the game. Each team, in its turn, places one of his flags in a point of the circumference that has not been used in a previous play. Each flag, once placed, cannot be moved. Once all $ 2n$ flags have been placed, territory is divided between the two teams. A point of the territory belongs to $ A$ if the closest flag to it is blue, and it belongs to $ B$ if the closest flag to it is white. If the closest blue flag to a point is at the same distance than the closest white flag to that point, the point is neutral (not from $ A$ nor from $ B$). A team wins the game is their points cover a greater area that that covered by the points of the other team. There is a draw if both cover equal areas. Prove that, for every $ n$, team $ B$ has a winning strategy.

Let $X=\{1,2,\dots,n\},$ and let $k\in X.$ Show that there are exactly $k\cdot n^{n-1}$ functions $f:X\to X$ such that for every $x\in X$ there is a $j\ge 0$ such that $f^{(j)}(x)\le k.$ [Here $f^{(j)}$ denotes the $j$th iterate of $f,$ so that $f^{(0)}(x)=x$ and $f^{(j+1)}(x)=f\left(f^{(j)}(x)\right).$]

Positive integers $x_1, x_2, \dots, x_n$ ($n \ge 4$) are arranged in a circle such that each $x_i$ divides the sum of the neighbors; that is \[ \frac{x_{i-1}+x_{i+1}}{x_i} = k_i \] is an integer for each $i$, where $x_0 = x_n$, $x_{n+1} = x_1$. Prove that \[ 2n \le k_1 + k_2 + \dots + k_n < 3n. \]

Find all functions $f: \mathbb{Z}\to \mathbb{Z}$ such that for all $m,n\in \mathbb{Z}$: \[f(m+f(n)) = f(m)+n.\]

Let $ \mathcal{P}$ be a square and let $ n$ be a nonzero positive integer for which we denote by $ f(n)$ the maximum number of elements of a partition of $ \mathcal{P}$ into rectangles such that each line which is parallel to some side of $ \mathcal{P}$ intersects at most $ n$ interiors (of rectangles). Prove that \[ 3 \cdot 2^{n\minus{}1} \minus{} 2 \le f(n) \le 3^n \minus{} 2.\]

Prove that there exists a set $ S$ of $ n \minus{} 2$ points inside a convex polygon $ P$ with $ n$ sides, such that any triangle determined by $3$ vertices of $ P$ contains exactly one point from $ S$ inside or on the boundaries.

Some checkers placed on an $n \times n$ checkerboard satisfy the following conditions: (a) every square that does not contain a checker shares a side with one that does; (b) given any pair of squares that contain checkers, there is a sequence of squares containing checkers, starting and ending with the given squares, such that every two consecutive squares of the sequence share a side. Prove that at least $(n^{2}-2)/3$ checkers have been placed on the board.

Find all functions $f: \mathbb{N} \rightarrow \mathbb{N}$ such that (a) $f(1)=1$ (b) $f(n+2)+(n^2+4n+3)f(n)=(2n+5)f(n+1)$ for all $n \in \mathbb{N}$. (c) $f(n)$ divides $f(m)$ if $m>n$.

In the space are given $2006$ distinct points, such that no $4$ of them are coplanar. One draws a segment between each pair of points. A natural number $m$ is called [i]good[/i] if one can put on each of these segments a positive integer not larger than $m$, so that every triangle whose three vertices are among the given points has the property that two of this triangle's sides have equal numbers put on, while the third has a larger number put on. Find the minimum value of a [i]good[/i] number $m$.

[b](a)[/b] Prove that for all positive integers $m,n$ we have \[\sum_{k=1}^n k(k+1)(k+2)\cdots (k+m-1)=\frac{n(n+1)(n+2) \cdots (n+m)}{m+1}\] [b](b)[/b] Let $P(x)$ be a polynomial with rational coefficients and degree $m.$ If $n$ tends to infinity, then prove that \[\frac{\sum_{k=1}^n P(k)}{n^{m+1}}\] Has a limit.

$\mathbb{N}_{10}$ is generalization of $\mathbb{N}$ that every hypernumber in $\mathbb{N}_{10}$ is something like: $\overline{...a_2a_1a_0}$ with $a_i \in {0,1..9}$ (Notice that $\overline {...000} \in \mathbb{N}_{10}$) Also we easily have $+,*$ in $\mathbb{N}_{10}$. first $k$ number of $a*b$= first $k$ nubmer of (first $k$ number of a * first $k$ number of b) first $k$ number of $a+b$= first $k$ nubmer of (first $k$ number of a + first $k$ number of b) Fore example $\overline {...999}+ \overline {...0001}= \overline {...000}$ Prove that every monic polynomial in $\mathbb{N}_{10}[x]$ with degree $d$ has at most $d^2$ roots.

Define a sequence $(a_n)_{n \geq 1}$ by $a_1 =1$ and $a_2 =2$ and $a_{n+2} = 2 a_{n+1} - a_n + 2$ for $n \geq 1$. prove that for any $m$ , $a_m a_{m+1}$ is also a term in this sequence.

The function $f:\mathbb R^{\ge 0} \longrightarrow \mathbb R^{\ge 0}$ satisfies the following properties for all $a,b\in \mathbb R^{\ge 0}$: [b]a)[/b] $f(a)=0 \Leftrightarrow a=0$ [b]b)[/b] $f(ab)=f(a)f(b)$ [b]c)[/b] $f(a+b)\le 2 \max \{f(a),f(b)\}$. Prove that for all $a,b\in \mathbb R^{\ge 0}$ we have $f(a+b)\le f(a)+f(b)$. [i]Proposed by Masoud Shafaei[/i]

a) Let $ n_{1},n_{2},\dots$ be a sequence of natural number such that $ n_{i}\geq2$ and $ \epsilon_{1},\epsilon_{2},\dots$ be a sequence such that $ \epsilon_{i}\in\{1,2\}$. Prove that the sequence: \[ \sqrt[n_{1}]{\epsilon_{1}\plus{}\sqrt[n_{2}]{\epsilon_{2}\plus{}\dots\plus{}\sqrt[n_{k}]{\epsilon_{k}}}}\]is convergent and its limit is in $ (1,2]$. Define $ \sqrt[n_{1}]{\epsilon_{1}\plus{}\sqrt[n_{2}]{\epsilon_{2}\plus{}\dots}}$ to be this limit. b) Prove that for each $ x\in(1,2]$ there exist sequences $ n_{1},n_{2},\dots\in\mathbb N$ and $ n_{i}\geq2$ and $ \epsilon_{1},\epsilon_{2},\dots$, such that $ n_{i}\geq2$ and $ \epsilon_{i}\in\{1,2\}$, and $ x\equal{}\sqrt[n_{1}]{\epsilon_{1}\plus{}\sqrt[n_{2}]{\epsilon_{2}\plus{}\dots}}$

Let $n$ be a positive integer and let $a_1, a_2, \ldots, a_n$ be positive reals. Show that $$\sum_{i=1}^{n} \frac{1}{2^i}(\frac{2}{1+a_i})^{2^i} \geq \frac{2}{1+a_1a_2\ldots a_n}-\frac{1}{2^n}.$$

Find all pairs of polynomials $p(x)$ and $q(x)$ with real coefficients for which \[p(x)q(x+1)-p(x+1)q(x)=1.\]

Suppose a function $f : \mathbb{Z}^+ \rightarrow \mathbb{Z}^+$ satisfies $f(f(n)) + f(n+1) = n+2$ for all positive integer $n$. Prove that $f(f(n)+n) = n+1$ for all positive integer $n$.