Let $ f(x) \equal{} \frac{x^2\plus{}1}{2x}$ for $ x \neq 0.$ Define $ f^{(0)}(x) \equal{} x$ and $ f^{(n)}(x) \equal{} f(f^{(n\minus{}1)}(x))$ for all positive integers $ n$ and $ x \neq 0.$ Prove that for all non-negative integers $ n$ and $ x \neq \{\minus{}1,0,1\}$ \[ \frac{f^{(n)}(x)}{f^{(n\plus{}1)}(x)} \equal{} 1 \plus{} \frac{1}{f \left( \left( \frac{x\plus{}1}{x\minus{}1} \right)^{2n} \right)}.\]

Let $ a,b,c$ be positive real numbers. Prove the inequality: $ \frac {a^3}{b^2} \plus{} \frac {b^3}{c^2} \plus{} \frac {c^3}{a^2}\ge \frac {a^2}{b} \plus{} \frac {b^2}{c} \plus{} \frac {c^2}{a}$

Suppose $\{a(n) \}_{n=1}^{\infty}$ is a sequence that: \[ a(n) =a(a(n-1))+a(n-a(n-1)) \ \ \ \forall \ n \geq 3\] and $a(1)=a(2)=1$. Prove that for each $n \geq 1$ , $a(2n) \leq 2a(n)$.

Find all strictly increasing functions $f: \mathbb{N}\to \mathbb{N}$ such that \[f(f(n))=3n.\]

Let $n\ge 1$ be an odd integer. Alice and Bob play the following game, taking alternating turns, with Alice playing first. The playing area consists of $n$ spaces, arranged in a line. Initially all spaces are empty. At each turn, a player either • places a stone in an empty space, or • removes a stone from a nonempty space $s,$ places a stone in the nearest empty space to the left of $s$ (if such a space exists), and places a stone in the nearest empty space to the right of $s$ (if such a space exists). Furthermore, a move is permitted only if the resulting position has not occurred previously in the game. A player loses if he or she is unable to move. Assuming that both players play optimally throughout the game, what moves may Alice make on her first turn?

The natural numbers are written in sequence, in increasing order, and by this we get an infinite sequence of digits. Find the least natural $k$, for which among the first $k$ digits of this sequence, any two nonzero digits have been written a different number of times. [i]Aleksandar Ivanov, Emil Kolev [/i]

Let $f(x)=\frac{ax+b}{cx+d}$ $F_n(x)=f(f(f...f(x)...))$ (with $n\ f's$) Suppose that $f(0) \not =0$, $f(f(0)) \not = 0$, and for some $n$ we have $F_n(0)=0$, show that $F_n(x)=x$ (for any valid x).

A sequence of numbers $a_n, n = 1,2, \ldots,$ is defined as follows: $a_1 = \frac{1}{2}$ and for each $n \geq 2$ \[ a_n = \frac{2 n - 3}{2 n} a_{n-1}. \] Prove that $\sum^n_{k=1} a_k < 1$ for all $n \geq 1.$

The integers from $1$ to $1993$ are written in a line in some order. The following operation is performed with this line: if the first number is $k$ then the first $k$ numbers are rewritten in reverse order. Prove that after some finite number of these operations, the first number in the line of numbers will be $1$.

Prove that for every positive integer $n$ there exists an $n$-digit number divisible by $5^n$ all of whose digits are odd.

Let $a_{1} < a_{2} < a_{3} < \cdots $ be an infinite increasing sequence of positive integers in which the number of prime factors of each term, counting repeated factors, is never more than $1987$. Prove that it is always possible to extract from $A$ an infinite subsequence $b_{1} < b_{2} < b_{3} < \cdots $ such that the greatest common divisor $(b_i, b_j)$ is the same number for every pair of its terms.

In a mathematical competition, some competitors are friends; friendship is mutual, that is, when $A$ is a friend of $B$, then $B$ is also a friend of $A$. We say that $n \geq 3$ different competitors $A_1, A_2, \ldots, A_n$ form a [i]weakly-friendly cycle [/i]if $A_i$ is not a friend of $A_{i+1}$ for $1 \leq i \leq n$ (where $A_{n+1} = A_1$), and there are no other pairs of non-friends among the components of the cycle. The following property is satisfied: "for every competitor $C$ and every weakly-friendly cycle $\mathcal{S}$ of competitors not including $C$, the set of competitors $D$ in $\mathcal{S}$ which are not friends of $C$ has at most one element" Prove that all competitors of this mathematical competition can be arranged into three rooms, such that every two competitors in the same room are friends. ([i]Serbia[/i])

Let $F_r=x^r\sin{rA}+y^r\sin{rB}+z^r\sin{rC}$, where $x,y,z,A,B,C$ are real and $A+B+C$ is an integral multiple of $\pi$. Prove that if $F_1=F_2=0$, then $F_r=0$ for all positive integral $r$.

A directed graph has each vertex with outdegree 2. Prove that it is possible to split the vertices into 3 sets so that for each vertex $v$, $v$ is not simultaneously in the same set with both of the vertices that it points to. [i]David Yang.[/i] [hide="Stronger Version"]See [url=http://www.artofproblemsolving.com/Forum/viewtopic.php?f=42&t=492100]here[/url].[/hide]

On each day of their tour of the West Indies, Sourav and Srinath have either an apple or an orange for breakfast. Sourav has oranges for the first $m$ days, apples for the next $m$ days, followed by oranges for the next $m$ days, and so on. Srinath has oranges for the first $n$ days, apples for the next $n$ days, followed by oranges for the next $n$ days, and so on. If $\gcd(m,n)=1$, and if the tour lasted for $mn$ days, on how many days did they eat the same kind of fruit?

Let $f_1(x)$ be a polynomial of degree $2$ with the leading coefficient positive and $f_{n+1}(x) =f_1(f_n(x))$ for $n\ge 1.$ Prove that if the equation $f_2(x)=0$ has four different non-positive real roots, then for arbitrary $n$ then $f_n(x)$ has $2^n$ different real roots.

For all natural $n$, an $n$-staircase is a figure consisting of unit squares, with one square in the first row, two squares in the second row, and so on, up to $n$ squares in the $n^{th}$ row, such that all the left-most squares in each row are aligned vertically. Let $f(n)$ denote the minimum number of square tiles requires to tile the $n$-staircase, where the side lengths of the square tiles can be any natural number. e.g. $f(2)=3$ and $f(4)=7$. (a) Find all $n$ such that $f(n)=n$. (b) Find all $n$ such that $f(n) = n+1$.

Let $S_0$ be a finite set of positive integers. We define finite sets $S_1, S_2, \cdots$ of positive integers as follows: the integer $a$ in $S_{n+1}$ if and only if exactly one of $a-1$ or $a$ is in $S_n$. Show that there exist infinitely many integers $N$ for which $S_N = S_0 \cup \{ N + a: a \in S_0 \}$.

Find all functions $f:\mathbb{N} \to \mathbb{N}$ such that for all $m,n\in \mathbb{N}$: [list][*] $f(2)=2$, [*] $f(mn)=f(m)f(n)$, [*] $f(n+1)>f(n)$. [/list]

Let $n$ be a positive integer, and $a_j$, for $j=1,2,\ldots,n$ are complex numbers. Suppose $I$ is an arbitrary nonempty subset of $\{1,2,\ldots,n\}$, the inequality $\left|-1+ \prod_{j\in I} (1+a_j) \right| \leq \frac 12$ always holds. Prove that $\sum_{j=1}^n |a_j| \leq 3$.

$n$ 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.

Determine all natural numbers $n$ such that for each natural number $a$ relatively prime with $n$ and $a \le 1 + \left\lfloor \sqrt{n} \right\rfloor$ there exists some integer $x$ with $a \equiv x^2 \mod n$. Remark: "Natural numbers" is the set of positive integers.

In a row are 23 boxes such that for $1\le k \le 23$, there is a box containing exactly $k$ balls. In one move, we can double the number of balls in any box by taking balls from another box which has more. Is it always possible to end up with exactly $k$ balls in the $k$-th box for $1\le k\le 23$?

Let $n$ be a positive integer. A $corner$ is a finite set $S$ of ordered $n$-tuples of positive integers such that if $a_1, a_2, \ldots, a_n, b_1, b_2, \ldots, b_n$ are positive integers with $a_k \geq b_k$ for $k = 1, 2, \ldots, n$ and $(a_1, a_2, \ldots, a_n) \in S$, then $(b_1, b_2, \ldots, b_n) \in S$. Prove that among any infinite collection of corners, there exist two corners, one of which is a subset of the other one.

Prove that every infinite sequence $S$ of distinct positive integers contains either an infinite subsequence such that for every pair of terms, neither term ever divides the other, or an infinite subsequence such that in every pair of terms, one always divides the other.