Let $f$ be a function with the following properties: 1) $f(n)$ is defined for every positive integer $n$; 2) $f(n)$ is an integer; 3) $f(2)=2$; 4) $f(mn)=f(m)f(n)$ for all $m$ and $n$; 5) $f(m)>f(n)$ whenever $m>n$. Prove that $f(n)=n$.

$S$ is the set of functions $f:\mathbb{N} \to \mathbb{R}$ that satisfy the following conditions: [b]I.[/b] $f(1) = 2$ [b]II.[/b] $f(n+1) \geq f(n) \geq \frac{n}{n + 1} f(2n)$ for $n = 1, 2, \ldots$ Find the smallest $M \in \mathbb{N}$ such that for any $f \in S$ and any $n \in \mathbb{N}, f(n) < M$.

Assume real numbers $a_i,b_i\,(i=0,1,\cdots,2n)$ satisfy the following conditions: (1) for $i=0,1,\cdots,2n-1$, we have $a_i+a_{i+1}\geq 0$; (2) for $j=0,1,\cdots,n-1$, we have $a_{2j+1}\leq 0$; (2) for any integer $p,q$, $0\leq p\leq q\leq n$, we have $\sum_{k=2p}^{2q}b_k>0$. Prove that $\sum_{i=0}^{2n}(-1)^i a_i b_i\geq 0$, and determine when the equality holds.

Two real sequence $ \{x_{n}\}$ and $ \{y_{n}\}$ satisfies following recurrence formula; $ x_{0}\equal{} 1$, $ y_{0}\equal{} 2007$ $ x_{n\plus{}1}\equal{} x_{n}\minus{}(x_{n}y_{n}\plus{}x_{n\plus{}1}y_{n\plus{}1}\minus{}2)(y_{n}\plus{}y_{n\plus{}1})$, $ y_{n\plus{}1}\equal{} y_{n}\minus{}(x_{n}y_{n}\plus{}x_{n\plus{}1}y_{n\plus{}1}\minus{}2)(x_{n}\plus{}x_{n\plus{}1})$ Then show that for all nonnegative integer $ n$, $ {x_{n}}^{2}\leq 2007$.

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 numbers $1, 2, \ldots, 2012$ are written on a blackboard. Each minute, a student goes up to the board, chooses two numbers $x$ and $y$, erases them, and writes the number $2x+2y$ on the board. This continues until only one number $N$ remains. Find the remainder when the maximum possible value of $N$ is divided by 1000. [i]Victor Wang.[/i]

There is a $2012\times 2012$ grid with rows numbered $1,2,\dots 2012$ and columns numbered $1,2,\dots, 2012$, and we place some rectangular napkins on it such that the sides of the napkins all lie on grid lines. Each napkin has a positive integer thickness. (in micrometers!) (a) Show that there exist $2012^2$ unique integers $a_{i,j}$ where $i,j \in [1,2012]$ such that for all $x,y\in [1,2012]$, the sum \[ \sum _{i=1}^{x} \sum_{j=1}^{y} a_{i,j} \] is equal to the sum of the thicknesses of all the napkins that cover the grid square in row $x$ and column $y$. (b) Show that if we use at most $500,000$ napkins, at least half of the $a_{i,j}$ will be $0$. [i]Proposed by Ray Li[/i]

Show that $\cos 1^{\circ}$ is irrational.

Find all surjective functions $f: \mathbb{N}\to \mathbb{N}$ such that for all $m,n\in \mathbb{N}$: \[m \vert n \Longleftrightarrow f(m) \vert f(n).\]

Define a sequence $ < a_n > _{n\geq0}$ by $ a_0 \equal{} 0$, $ a_1 \equal{} 1$ and \[ a_n \equal{} 2a_{n \minus{} 1} \plus{} a_{n \minus{} 2},\] for $ n\geq2.$ $ (a)$ For every $ m > 0$ and $ 0\leq j\leq m,$ prove that $ 2a_m$ divides $ a_{m \plus{} j} \plus{} ( \minus{} 1)^ja_{m \minus{} j}$. $ (b)$ Suppose $ 2^k$ divides $ n$ for some natural numbers $ n$ and $ k$. Prove that $ 2^k$ divides $ a_n.$

There are 100 apples on the table with total weight of 10 kg. Each apple weighs no less than 25 grams. The apples need to be cut for 100 children so that each of the children gets 100 grams. Prove that you can do it in such a way that each piece weighs no less than 25 grams.

Show that every positive rational number can be written as a quotient of products of factorials of (not necessarily distinct) primes. For example, $ \frac{10}9\equal{}\frac{2!\cdot 5!}{3!\cdot 3!\cdot 3!}.$

Consider a regular $n$-gon with $n$ odd. Given two adjacent vertices $A_{1}$ and $A_{2},$ define the sequence $(A_{k})$ of vertices of the $n$-gon as follows: For $k\ge 3,\, A_{k}$ is the vertex lying on the perpendicular bisector of $A_{k-2}A_{k-1}.$ Find all $n$ for which each vertex of the $n$-gon occurs in this sequence.

Show that, for any fixed integer $\,n \geq 1,\,$ the sequence \[ 2, \; 2^2, \; 2^{2^2}, \; 2^{2^{2^2}}, \ldots (\mbox{mod} \; n) \] is eventually constant. [The tower of exponents is defined by $a_1 = 2, \; a_{i+1} = 2^{a_i}$. Also $a_i \; (\mbox{mod} \; n)$ means the remainder which results from dividing $a_i$ by $n$.]

Consider a positive integer $a > 1$. If $a$ is not a perfect square then at the next move we add $3$ to it and if it is a perfect square we take the square root of it. Define the trajectory of a number $a$ as the set obtained by performing this operation on $a$. For example the cardinality of $3$ is $\{3, 6, 9\}$. Find all $n$ such that the cardinality of $n$ is finite. The following part problems may attract partial credit. $\textbf{(a)}$Show that the cardinality of the trajectory of a number cannot be $1$ or $2$. $\textbf{(b)}$Show that $\{3, 6, 9\}$ is the only trajectory with cardinality $3$. $\textbf{(c)}$ Show that there for all $k \geq 3$, there exists a number such that the cardinality of its trajectory is $k$. $\textbf{(d)}$ Give an example of a number with cardinality of trajectory as infinity.

Let $f$ be a function with the following properties: 1) $f(n)$ is defined for every positive integer $n$; 2) $f(n)$ is an integer; 3) $f(2)=2$; 4) $f(mn)=f(m)f(n)$ for all $m$ and $n$; 5) $f(m)>f(n)$ whenever $m>n$. Prove that $f(n)=n$.

Assume real numbers $a_i,b_i\,(i=0,1,\cdots,2n)$ satisfy the following conditions: (1) for $i=0,1,\cdots,2n-1$, we have $a_i+a_{i+1}\geq 0$; (2) for $j=0,1,\cdots,n-1$, we have $a_{2j+1}\leq 0$; (2) for any integer $p,q$, $0\leq p\leq q\leq n$, we have $\sum_{k=2p}^{2q}b_k>0$. Prove that $\sum_{i=0}^{2n}(-1)^i a_i b_i\geq 0$, and determine when the equality holds.

Let $k$ be a positive integer. At the European Chess Cup every pair of players played a game in which somebody won (there were no draws). For any $k$ players there was a player against whom they all lost, and the number of players was the least possible for such $k$. Is it possible that at the Closing Ceremony all the participants were seated at the round table in such a way that every participant was seated next to both a person he won against and a person he lost against. [i]Proposed by Matija Bucić.[/i]

For any positive integer $n$ let $f(n)$ be the number of divisors of $n$ ending with $1$ or $9$ in base $10$ and let $g(n)$ be the number of divisors of $n$ ending with digit $3$ or $7$ in base $10$. Prove that $f(n)\geqslant g(n)$ for all nonnegative integers $n$. [i](Swiss Mathematical Olympiad 2011, Final round, problem 9)[/i]

The function $f : \mathbb N \to \mathbb R$ satisfies $f(1) = 1, f(2) = 2$ and \[f (n+2) = f(n+2 - f(n+1) ) + f(n+1 - f(n) ).\] Show that $0 \leq f(n+1) - f(n) \leq 1$. Find all $n$ for which $f(n) = 1025$.

Show that, for any fixed integer $\,n \geq 1,\,$ the sequence \[ 2, \; 2^2, \; 2^{2^2}, \; 2^{2^{2^2}}, \ldots (\mbox{mod} \; n) \] is eventually constant. [The tower of exponents is defined by $a_1 = 2, \; a_{i+1} = 2^{a_i}$. Also $a_i \; (\mbox{mod} \; n)$ means the remainder which results from dividing $a_i$ by $n$.]

Let $\alpha $ be an angle such that $\cos \alpha = \frac pq$, where $p$ and $q$ are two integers. Prove that the number $q^n \cos n \alpha$ is an integer.

Let $p(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$ be a monic polynomial of degree $n>2$, with real coefficients and all its roots real and different from zero. Prove that for all $k=0,1,2,\cdots,n-2$, at least one of the coefficients $a_k,a_{k+1}$ is different from zero.

Let $ x,y$ be real numbers such that the numbers $ x\plus{}y, x^2\plus{}y^2, x^3\plus{}y^3$ and $ x^4\plus{}y^4$ are integers. Prove that for all positive integers $ n$, the number $ x^n \plus{} y^n$ is an integer.

Let's say we have a [i]nice[/i] representation of the positive integer $ n$ if we write it as a sum of powers of 2 in such a way that there are at most two equal powers in the sum (representations differing only in the order of their summands are considered to be the same). a) Write down the 5 nice representations of 10. b) Find all positive integers with an even number of nice representations.