A convex polygon is such that the distance between any two vertices does not exceed $ 1$. $ (i)$ Prove that the distance between any two points on the boundary of the polygon does not exceed $ 1$. $ (ii)$ If $ X$ and $ Y$ are two distinct points inside the polygon, prove that there exists a point $ Z$ on the boundary of the polygon such that $ XZ \plus{} YZ\le1$.

For any positive integer $ x$ define $ g(x)$ as greatest odd divisor of $ x,$ and \[ f(x) \equal{} \begin{cases} \frac {x}{2} \plus{} \frac {x}{g(x)} & \text{if \ \(x\) is even}, \\ 2^{\frac {x \plus{} 1}{2}} & \text{if \ \(x\) is odd}. \end{cases} \] Construct the sequence $ x_1 \equal{} 1, x_{n \plus{} 1} \equal{} f(x_n).$ Show that the number 1992 appears in this sequence, determine the least $ n$ such that $ x_n \equal{} 1992,$ and determine whether $ n$ is unique.

Let $ c$ be a positive integer. The sequence $ a_1,a_2,\ldots$ is defined as follows $ a_1\equal{}c$, $ a_{n\plus{}1}\equal{}a_n^2\plus{}a_n\plus{}c^3$ for all positive integers $ n$. Find all $ c$ so that there are integers $ k\ge1$ and $ m\ge2$ so that $ a_k^2\plus{}c^3$ is the $ m$th power of some integer.

A $(2^n - 1) \times (2^n +1)$ board is to be divided into rectangles with sides parallel to the sides of the board and integer side lengths such that the area of each rectangle is a power of 2. Find the minimum number of rectangles that the board may be divided into.

Let $ x_0 \equal{} 1$ and for $ n\ge0,$ let $ x_{n \plus{} 1} \equal{} 3x_n \plus{} \left\lfloor x_n\sqrt {5}\right\rfloor.$ In particular, $ x_1 \equal{} 5,\ x_2 \equal{} 26,\ x_3 \equal{} 136,\ x_4 \equal{} 712.$ Find a closed-form expression for $ x_{2007}.$ ($ \lfloor a\rfloor$ means the largest integer $ \le a.$)

Define the sequence $(x_n)$ by $x_0 = 0$ and for all $n \in \mathbb N,$ \[x_n=\begin{cases} x_{n-1} + (3^r - 1)/2,&\mbox{ if } n = 3^{r-1}(3k + 1);\\ x_{n-1} - (3^r + 1)/2, & \mbox{ if } n = 3^{r-1}(3k + 2).\end{cases}\] where $k \in \mathbb N_0, r \in \mathbb N$. Prove that every integer occurs in this sequence exactly once.

Prove that every positive integer can be represented in the form \[3^{u_1} \ldots 2^{v_1} + 3^{u_2} \ldots 2^{v_2} + \ldots + 3^{u_k} \ldots 2^{v_k}\] with integers $u_1, u_2, \ldots , u_k, v_1, \ldots, v_k$ such that $u_1 > u_2 >\ldots > u_k\ge 0$ and $0 \le v_1 < v_2 <\ldots < v_k$.

Suppose that a finite group has exactly $ n$ elements of order $ p,$ where $ p$ is a prime. Prove that either $ n\equal{}0$ or $ p$ divides $ n\plus{}1.$

The sequences $(a_n),(b_n)$ are defined by $a_0=1,b_0=4$ and for $n\ge 0$ \[a_{n+1}=a_n^{2001}+b_n,\ \ b_{n+1}=b_n^{2001}+a_n\] Show that $2003$ is not divisor of any of the terms in these two sequences.

For every natural number $n$ we define $f(n)$ by the following rule: $f(1) = 1$ and for $n>1$ then $f(n) = 1 + a_1 \cdot p_1 + \ldots + a_k \cdot p_k$, where $n = p_1^{a_1} \cdots p_k^{a_k}$ is the canonical prime factorisation of $n$ ($p_1, \ldots, p_k$ are distinct primes and $a_1, \ldots, a_k$ are positive integers). For every positive integer $s$, let $f_s(n) = f(f(\ldots f(n))\ldots)$, where on the right hand side there are exactly $s$ symbols $f$. Show that for every given natural number $a$, there is a natural number $s_0$ such that for all $s > s_0$, the sum $f_s(a) + f_{s-1}(a)$ does not depend on $s$.

Find functions $ f: \mathbb{N} \rightarrow \mathbb{N}$, such that $ f(x^2 \plus{} f(y)) \equal{} xf(x) \plus{} y$, for $ x,y \in \mathbb{N}$.

Prove that for every positive integer $ n$, $ n^n \le (n!)^2 \le \left( \frac{(n\plus{}1)(n\plus{}2)}{6} \right) ^n.$

Let $A$ be the set of $n$-digit integers whose digits are all from $\{ 1, 2, 3, 4, 5 \}$. $B$ is subset of $A$ such that it contains digit $5$, and there is no digit $3$ in front of digit $5$ (i.e. for $n = 2$, $35$ is not allowed, but $53$ is allowed). How many elements does set $B$ have?

[b]Problem 1. [/b]In the cells of square table are written the numbers $1$, $0$ or $-1$ so that in every line there is exactly one $1$, amd exactly one $-1$. Each turn we change the places of two columns or two rows. Is it possible, from any such table, after finite number of turns to obtain its opposite table (two tables are opposite if the sum of the numbers written in any two corresponding squares is zero)? [i] Emil Kolev[/i]

Is there a sequence $ a_1,a_2,\ldots$ of positive reals satisfying simoultaneously the following inequalities for all positive integers $ n$: a) $ a_1\plus{}a_2\plus{}\ldots\plus{}a_n\le n^2$ b) $ \frac1{a_1}\plus{}\frac1{a_2}\plus{}\ldots\plus{}\frac1{a_n}\le2008$?

For positive integer $k>1$, let $f(k)$ be the number of ways of factoring $k$ into product of positive integers greater than $1$ (The order of factors are not countered, for example $f(12)=4$, as $12$ can be factored in these $4$ ways: $12,2\cdot 6,3\cdot 4, 2\cdot 2\cdot 3$. Prove: If $n$ is a positive integer greater than $1$, $p$ is a prime factor of $n$, then $f(n)\leq \frac{n}{p}$

In Happy City there are $2014$ citizens called $A_1, A_2, \dots , A_{2014}$. Each of them is either [i]happy[/i] or [i]unhappy[/i] at any moment in time. The mood of any citizen $A$ changes (from being unhappy to being happy or vice versa) if and only if some other happy citizen smiles at $A$. On Monday morning there were $N$ happy citizens in the city. The following happened on Monday during the day: the citizen $A_1$ smiled at citizen $A_2$, then $A_2$ smiled at $A_3$, etc., and, finally, $A_{2013}$ smiled at $A_{2014}$. Nobody smiled at anyone else apart from this. Exactly the same repeated on Tuesday, Wednesday and Thursday. There were exactly $2000$ happy citizens on Thursday evening. Determine the largest possible value of $N$.

The sequence $a_n$ is defined by $a_0=a_1=1$ and $a_{n+1}=14a_n-a_{n-1}-4$,for all positive integers $n$. Prove that all terms of this sequence are perfect squares.

Let $n$ be a fixed positive integer. Initially, $n$ 1's are written on a blackboard. Every minute, David picks two numbers $x$ and $y$ written on the blackboard, erases them, and writes the number $(x+y)^4$ on the blackboard. Show that after $n-1$ minutes, the number written on the blackboard is at least $2^{\frac{4n^2-4}{3}}$. [i]Proposed by Calvin Deng[/i]

The sequence $ (a_n)$ satisfies $ a_1 \equal{} 1$ and $ \displaystyle 5^{(a_{n\plus{}1}\minus{}a_n)} \minus{} 1 \equal{} \frac{1}{n\plus{}\frac{2}{3}}$ for $ n \geq 1$. Let $ k$ be the least integer greater than $ 1$ for which $ a_k$ is an integer. Find $ k$.

In boxes labeled $0$, $1$, $2$, $\dots$, we place integers according to the following rules: $\bullet$ If $p$ is a prime number, we place it in box $1$. $\bullet$ If $a$ is placed in box $m_a$ and $b$ is placed in box $m_b$, then $ab$ is placed in the box labeled $am_b+bm_a$. Find all positive integers $n$ that are placed in the box labeled $n$.

If $a_{1}$, $a_{2}$, $\ldots$, $a_{n}\geq 0$ are such that \[a_{1}^{2}+\cdots+a_{n}^{2}=1,\] then find the maximum value of the product $(1-a_{1})\cdots (1-a_{n})$.

The sequence $ (a_n)$ satisfies $ a_0 \equal{} 0$ and $ \displaystyle a_{n \plus{} 1} \equal{} \frac85a_n \plus{} \frac65\sqrt {4^n \minus{} a_n^2}$ for $ n\ge0$. Find the greatest integer less than or equal to $ a_{10}$.

A black pawn and a white pawn are placed on the first square and the last square of a $ 1\times n$ chessboard, respectively. Wiwit and Siti move alternatingly. Wiwit has the white pawn, and Siti has the black pawn. The white pawn moves first. In every move, the player moves her pawn one or two squares to the right or to the left, without passing the opponent's pawn. The player who cannot move anymore loses the game. Which player has the winning strategy? Explain the strategy.

a)Prove that $\frac{1}{2}+\frac{1}{3}+...+\frac{1}{{{2}^{m}}}<m$, for any $m\in {{\mathbb{N}}^{*}}$. b)Let ${{p}_{1}},{{p}_{2}},...,{{p}_{n}}$ be the prime numbers less than ${{2}^{100}}$. Prove that $\frac{1}{{{p}_{1}}}+\frac{1}{{{p}_{2}}}+...+\frac{1}{{{p}_{n}}}<10$