Define a [i]beautiful number[/i] to be an integer of the form $a^n$, where $a\in\{3,4,5,6\}$ and $n$ is a positive integer. Prove that each integer greater than $2$ can be expressed as the sum of pairwise distinct beautiful numbers. [i]Proposed by Matthew Babbitt[/i]

Let $m$ and $n$ be positive integers. Mr. Fat has a set $S$ containing every rectangular tile with integer side lengths and area of a power of $2$. Mr. Fat also has a rectangle $R$ with dimensions $2^m \times 2^n$ and a $1 \times 1$ square removed from one of the corners. Mr. Fat wants to choose $m + n$ rectangles from $S$, with respective areas $2^0, 2^1, \ldots, 2^{m + n - 1}$, and then tile $R$ with the chosen rectangles. Prove that this can be done in at most $(m + n)!$ ways. [i]Palmer Mebane.[/i]

Given integer $a_1\geq 2$. For integer $n\geq 2$, define $a_n$ to be the smallest positive integer which is not coprime to $a_{n-1}$ and not equal to $a_1,a_2,\cdots, a_{n-1}$. Prove that every positive integer except 1 appears in this sequence $\{a_n\}$.

Let $M=\{1,2,\ldots,3 \cdot n\}$. Partition $M$ into three sets $A,B,C$ which $card$ $A$ $=$ $card$ $B$ $=$ $card$ $C$ $=$ $n .$ Prove that there exists $a$ in $A,b$ in $B, c$ in $C$ such that or $a=b+c,$ or $b=c+a,$ or $c=a+b$ [i]Edited by orl.[/i]

For positive integers $a$ and $k$, define the sequence $a_1,a_2,\ldots$ by \[a_1=a,\qquad\text{and}\qquad a_{n+1}=a_n+k\cdot\varrho(a_n)\qquad\text{for } n=1,2,\ldots\] where $\varrho(m)$ denotes the product of the decimal digits of $m$ (for example, $\varrho(413)=12$ and $\varrho(308)=0$). Prove that there are positive integers $a$ and $k$ for which the sequence $a_1,a_2,\ldots$ contains exactly $2009$ different numbers.

In Greifswald there are three schools called $A,B$ and $C$, each of which is attended by at least one student. Among any three students, one from $A$, one from $B$ and one from $C$, there are two knowing each other and two not knowing each other. Prove that at least one of the following holds: [list] [*]Some student from $A$ knows all students from $B$. [*]Some student from $B$ knows all students from $C$. [*] Some student from $C$ knows all students from $A$.[/list]

Find all functions $f$ from positive integers to themselves such that: 1)$f(mn)=f(m)f(n)$ for all positive integers $m, n$ 2)$\{1, 2, ..., n\}=\{f(1), f(2), ... f(n)\}$ is true for infinitely many positive integers $n$.

Let $P(x)$ be a nonzero polynomial with integer coefficients. Let $a_{0}=0$ and for $i \ge 0$ define $a_{i+1}=P(a_{i})$. Show that $\gcd ( a_{m}, a_{n})=a_{ \gcd (m, n)}$ for all $m, n \in \mathbb{N}$.

Let $n\ge 2$ be a given integer. Show that the number of strings of length $n$ consisting of $0'$s and $1'$s such that there are equal number of $00$ and $11$ blocks in each string is equal to \[2\binom{n-2}{\left \lfloor \frac{n-2}{2}\right \rfloor}\]

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 $\frac{1}{{{p}_{1}}}+\frac{1}{{{p}_{2}}}+...+\frac{1}{{{p}_{n}}}<10$

Prove that for every given positive integer $k$, there exist infinitely many $n$, such that $2^{n}+3^{n}-1, 2^{n}+3^{n}-2,\ldots, 2^{n}+3^{n}-k$ are all composite numbers.

Let $M_n$ be the $n\times n$ matrix with entries as follows: for $1\leq i \leq n$, $m_{i,i}=10$; for $1\leq i \leq n-1, m_{i+1,i}=m_{i,i+1}=3$; all other entries in $M_n$ are zero. Let $D_n$ be the determinant of matrix $M_n$. Then $\displaystyle \sum_{n=1}^{\infty} \dfrac{1}{8D_n+1}$ can be represented as $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$. Note: The determinant of the $1\times 1$ matrix $[a]$ is $a$, and the determinant of the $2\times 2$ matrix $\left[ \begin{array}{cc} a & b \\ c & d \end{array} \right]=ad-bc$; for $n\geq 2$, the determinant of an $n\times n$ matrix with first row or first column $a_1\ a_2\ a_3 \dots\ a_n$ is equal to $a_1C_1 - a_2C_2 + a_3C_3 - \dots + (-1)^{n+1} a_nC_n$, where $C_i$ is the determinant of the $(n-1)\times (n-1)$ matrix found by eliminating the row and column containing $a_i$.

Let $a_{1}, a_{2}, \cdots, a_{k}$ be relatively prime positive integers. Determine the largest integer which cannot be expressed in the form \[x_{1}a_{2}a_{3}\cdots a_{k}+x_{2}a_{1}a_{3}\cdots a_{k}+\cdots+x_{k}a_{1}a_{2}\cdots a_{k-1}\] for some nonnegative integers $x_{1}, x_{2}, \cdots, x_{k}$.

If \[ \sum_{n=1}^{\infty}\frac{\frac11 + \frac12 + \dots + \frac 1n}{\binom{n+100}{100}} = \frac pq \] for relatively prime positive integers $p,q$, find $p+q$. [i]Proposed by Michael Kural[/i]

Consider the following one-person game: A player starts with score $0$ and writes the number $20$ on an empty whiteboard. At each step, she may erase any one integer (call it a) and writes two positive integers (call them $b$ and $c$) such that $b + c = a$. The player then adds $b\times c$ to her score. She repeats the step several times until she ends up with all $1$'s on the whiteboard. Then the game is over, and the final score is calculated. Let $M, m$ be the maximum and minimum final score that can be possibly obtained respectively. Find $M-m$.