Consider the sequence $ a_1\equal{}\frac{3}{2}, a_{n\plus{}1}\equal{}\frac{3a_n^2\plus{}4a_n\minus{}3}{4a_n^2}.$ $ (a)$ Prove that $ 1<a_n$ and $ a_{n\plus{}1}<a_n$ for all $ n$. $ (b)$ From $ (a)$ it follows that $ \displaystyle\lim_{n\to\infty}a_n$ exists. Find this limit. $ (c)$ Determine $ \displaystyle\lim_{n\to\infty}a_1a_2a_3...a_n$.

Find all positive integers $m$ such that there exist positive integers $a_1,a_2,\ldots,a_{1378}$ such that: \[ m=\sum_{k=1}^{1378}{\frac{k}{a_k}}. \]

A computer screen shows a $98 \times 98$ chessboard, colored in the usual way. One can select with a mouse any rectangle with sides on the lines of the chessboard and click the mouse button: as a result, the colors in the selected rectangle switch (black becomes white, white becomes black). Find, with proof, the minimum number of mouse clicks needed to make the chessboard all one color.

Given an integer $k\ge 2$. Prove that there exist $k$ pairwise distinct positive integers $a_1,a_2,\ldots,a_k$ such that for any non-negative integers $b_1,b_2,\ldots,b_k,c_1,c_2,\ldots,c_k$ satisfying $a_1\le b_i\le 2a_i, i=1,2,\ldots,k$ and $\prod_{i=1}^{k}b_i^{c_i}<\prod_{i=1}^{k}b_i$, we have \[k\prod_{i=1}^{k}b_i^{c_i}<\prod_{i=1}^{k}b_i.\]

For a finite non empty set of primes $P$, let $m(P)$ denote the largest possible number of consecutive positive integers, each of which is divisible by at least one member of $P$. (i) Show that $|P|\le m(P)$, with equality if and only if $\min(P)>|P|$. (ii) Show that $m(P)<(|P|+1)(2^{|P|}-1)$. (The number $|P|$ is the size of set $P$) [i]Dan Schwarz, Romania[/i]

Let $a,b,c\ge0$. Show that $(a^2+2bc)^{2012}+(b^2+2ca)^{2012}+(c^2+2ab)^{2012}\le (a^2+b^2+c^2)^{2012}+2(ab+bc+ca)^{2012}$. [i]Calvin Deng.[/i]

Is $\cos 1^\circ$ rational? Prove.

Let $k$ be a positive integer. Prove that one can partition the set $\{ 0,1,2,3, \cdots ,2^{k+1}-1 \}$ into two disdinct subsets $\{ x_1,x_2, \cdots, x_{2k} \}$ and $\{ y_1, y_2, \cdots, y_{2k} \}$ such that $\sum_{i=1}^{2^k} x_i^m =\sum_{i=1}^{2^k} y_i^m$ for all $m \in \{ 1,2, \cdots, k \}$.

For any $ n\ge 2 $ natural, show that the following inequality holds: $$ \sum_{i=2}^n\frac{1}{\sqrt[i]{(2i)!}}\ge\frac{n-1}{2n+2} . $$

Fix an integer $k>2$. Two players, called Ana and Banana, play the following game of numbers. Initially, some integer $n \ge k$ gets written on the blackboard. Then they take moves in turn, with Ana beginning. A player making a move erases the number $m$ just written on the blackboard and replaces it by some number $m'$ with $k \le m' < m$ that is coprime to $m$. The first player who cannot move anymore loses. An integer $n \ge k $ is called good if Banana has a winning strategy when the initial number is $n$, and bad otherwise. Consider two integers $n,n' \ge k$ with the property that each prime number $p \le k$ divides $n$ if and only if it divides $n'$. Prove that either both $n$ and $n'$ are good or both are bad.

An integer sequence $\{a_{n}\}_{n \ge 1}$ is defined by \[a_{1}=1, \; a_{n+1}=a_{n}+\lfloor \sqrt{a_{n}}\rfloor.\] Show that $a_{n}$ is a square if and only if $n=2^{k}+k-2$ for some $k \in \mathbb{N}$.

Let $ k$ be an integer, $ k \geq 2$, and let $ p_{1},\ p_{2},\ \ldots,\ p_{k}$ be positive reals with $ p_{1} \plus{} p_{2} \plus{} \ldots \plus{} p_{k} \equal{} 1$. Suppose we have a collection $ \left(A_{1,1},\ A_{1,2},\ \ldots,\ A_{1,k}\right)$, $ \left(A_{2,1},\ A_{2,2},\ \ldots,\ A_{2,k}\right)$, $ \ldots$, $ \left(A_{m,1},\ A_{1,2},\ \ldots,\ A_{m,k}\right)$ of $ k$-tuples of finite sets satisfying the following two properties: (i) for every $ i$ and every $ j \neq j^{\prime}$, $ A_{i,j}\cap A_{i,j^{\prime}} \equal{} \emptyset$, and (ii) for every $ i\neq i^{\prime}$ there exist $ j\neq j^{\prime}$ for which $ A_{i,j} \cap A_{i^{\prime},j^{\prime}}\neq\emptyset$. Prove that \[ \sum_{b \equal{} 1}^{m}{\prod_{a \equal{} 1}^{k}{p_{a}^{|A_{b,a}|}}} \leq 1. \]

There are $n$ cities, $2$ airline companies in a country. Between any two cities, there is exactly one $2$-way flight connecting them which is operated by one of the two companies. A female mathematician plans a travel route, so that it starts and ends at the same city, passes through at least two other cities, and each city in the route is visited once. She finds out that wherever she starts and whatever route she chooses, she must take flights of both companies. Find the maximum value of $n$.

The sequence $a_i$ is defined as $a_1 = 2, a_2 = 3$, and $a_{n+1} = 2a_{n-1}$ or $a_{n+1} = 3a_n - 2a_{n-1}$ for all integers $n \ge 2$. Prove that no term in $a_i$ is in the range $[1612, 2012]$.

Let $a$ and $b$ be positive integers greater than $2$. Prove that there exists a positive integer $k$ and a finite sequence $n_1$, $\cdots$, $n_k$ of positive integers such that $n_1 =a$, $n_k =b$, and $n_i n_{i+1}$ is divisible by $n_{i}+n_{i+1}$ for each $i$ $(1 \le i \le k)$.

The eleven members of a cricket team are numbered $1,2,...,11$. In how many ways can the entire cricket team sit on the eleven chairs arranged around a circular table so that the numbers of any two adjacent players differ by one or two ?

Find all numbers $N=\overline{a_1a_2\ldots a_n}$ for which $9\times \overline{a_1a_2\ldots a_n}=\overline{a_n\ldots a_2a_1}$ such that at most one of the digits $a_1,a_2,\ldots ,a_n$ is zero.

Let $ f(x)\equal{}\sum_{n\equal{}1}^{\infty} a_n/(x\plus{}n^2), \;(x \geq 0)\ ,$ where $ \sum_{n\equal{}1}^{\infty} |a_n|n^{\minus{} \alpha} < \infty$ for some $ \alpha > 2$. Let us assume that for some $ \beta > 1/{\alpha}$, we have $ f(x)\equal{}O(e^{\minus{}x^{\beta}})$ as $ x \rightarrow \infty$. Prove that $ a_n$ is identically $ 0$. [i]G. Halasz[/i]

Let $N$ be a positive integer. Alberto and Barbara write numbers on a blackboard taking turns, according to the following rules. Alberto starts writing $1$, and thereafter if a player has written $n$ on a certain move, his adversary is allowed to write $n+1$ or $2n$ as long as he/she does not obtain a number greater than $N$. The player who writes $N$ wins. $(a)$ Determine which player has a winning strategy for $N=2005$. $(b)$ Determine which player has a winning strategy for $N=2004$. $(c)$ Find for how many integers $N\le 2005$ Barbara has a winning strategy.

Let $\mathbb{R}^{+}$ denote the set of all positive real numbers. Find all functions $f:\mathbb{R}^{+}\longrightarrow \mathbb{R}$ satisfying \[f(x)+f(y)\le \frac{f(x+y)}{2}, \frac{f(x)}{x}+\frac{f(y)}{y}\ge \frac{f(x+y)}{x+y},\] for all $x, y\in \mathbb{R}^{+}$.

$ n\geq 2$ cars are participating in a rally. The cars leave the start line at different times and arrive at the finish line at different times. During the entire rally each car takes over any other car at most once , the number of cars taken over by each car is different and each car is taken over by the same number of cars. Find all possible values of $ n$

Determine all strictly increasing functions $f: \mathbb{N}\to\mathbb{N}$ satisfying $nf(f(n))=f(n)^2$ for all positive integers $n$. [i]Carl Lian and Brian Hamrick.[/i]

Given an integer $k\ge 2$. Prove that there exist $k$ pairwise distinct positive integers $a_1,a_2,\ldots,a_k$ such that for any non-negative integers $b_1,b_2,\ldots,b_k,c_1,c_2,\ldots,c_k$ satisfying $a_1\le b_i\le 2a_i, i=1,2,\ldots,k$ and $\prod_{i=1}^{k}b_i^{c_i}<\prod_{i=1}^{k}b_i$, we have \[k\prod_{i=1}^{k}b_i^{c_i}<\prod_{i=1}^{k}b_i.\]

Every non-negative integer is coloured white or red, so that: • there are at least a white number and a red number; • the sum of a white number and a red number is white; • the product of a white number and a red number is red. Prove that the product of two red numbers is always a red number, and the sum of two red numbers is always a red number.

Prove $ \frac{1}{10\sqrt2}<\frac{1}{2}\frac{3}{4}\frac{5}{6}...\frac{99}{100}<\frac{1}{10} $