Let $n$ be an integer greater than $1$. Define \[x_1 = n, y_1 = 1, x_{i+1} =\left[ \frac{x_i+y_i}{2}\right] , y_{i+1} = \left[ \frac{n}{x_{i+1}}\right], \qquad \text{for }i = 1, 2, \ldots\ ,\] where $[z]$ denotes the largest integer less than or equal to $z$. Prove that \[ \min \{x_1, x_2, \ldots, x_n \} =[ \sqrt n ]\]

Given $k \in \mathbb{N}^+$. A sequence of subset of the integer set $\mathbb{Z} \supseteq I_1 \supseteq I_2 \supseteq \cdots \supseteq I_k$ is called a $k-chain$ if for each $1 \le i \le k$ we have (i) $168 \in I_i$; (ii) $\forall x, y \in I_i$, we have $x-y \in I_i$. Determine the number of $k-chain$ in total.

Let $(b_n)_{n \ge 0}=\sum_{k=0}^{n} (a_0+kd)$ for positive integers $a_0$ and $d$. We consider all such sequences containing an element $b_i$ which equals $2010$. Determine the greatest possible value of $i$ and for this value the integers $a_0$ and $d$. [i](41th Austrian Mathematical Olympiad, regional competition, problem 4)[/i]

Call a sequence of positive integers $\{a_n\}$ good if for any distinct positive integers $m,n$, one has $$\gcd(m,n) \mid a_m^2 + a_n^2 \text{ and } \gcd(a_m,a_n) \mid m^2 + n^2.$$ Call a positive integer $a$ to be $k$-good if there exists a good sequence such that $a_k = a$. Does there exists a $k$ such that there are exactly $2019$ $k$-good positive integers?

Let the sequence $a_i$ be defined as $a_{i+1} = 2^{a_i}$. Find the number of integers $1 \le n \le 1000$ such that if $a_0 = n$, then $100$ divides $a_{1000} - a_1$.

Let a sequence $(x_n)$ be given and let $y_n = x_{n-1} +2 x_n $ for $n>1.$ Suppose that the sequence $(y_n)$ converges. Prove that the sequence $(x_n)$ converges, too.

Suppose a sequence of reals $\{a_n\}_{n\geq 0}$ satisfies $a_0 = 0$, $\frac{100}{101} <a_{100}<1$, and $$2a_n - a_{n-1} -a_{n+1} \leq 2 (1-a_n )^3$$ for every $n\geq 1$. (1) Define a sequence $b_n = a_n - \frac{n}{n+1}$. Prove that $b_n\leq b_{n+1}$ for any $n\geq 100$. (2) Determine whether infinite series $\sum_{n=1}^\infty \frac{a_n}{n^2}$ converges or diverges.

Given a sequence of positive integers $a_1, a_2, a_3, \ldots$ such that for any positive integers $k$, $l$ we have $k+l ~ | ~ a_k + a_l$. Prove that for all positive integers $k > l$, $a_k - a_l$ is divisible by $k-l$.

If the distinct non-zero numbers $x ( y - z),~ y(z - x),~ z(x - y )$ form a geometric progression with common ratio $r$, then $r$ satisfies the equation $\textbf{(A) }r^2+r+1=0\qquad\textbf{(B) }r^2-r+1=0\qquad\textbf{(C) }r^4+r^2-1=0$ $\qquad\textbf{(D) }(r+1)^4+r=0\qquad \textbf{(E) }(r-1)^4+r=0$

Let $x_n = \sqrt[2]{2+\sqrt[3]{3+\cdots+\sqrt[n]{n}}}.$ Prove that \[x_{n+1}-x_n <\frac{1}{n!} \quad n=2,3,\cdots\]

The $n$ contestant of EGMO are named $C_1, C_2, \cdots C_n$. After the competition, they queue in front of the restaurant according to the following rules. [list] [*]The Jury chooses the initial order of the contestants in the queue. [*]Every minute, the Jury chooses an integer $i$ with $1 \leq i \leq n$. [list] [*]If contestant $C_i$ has at least $i$ other contestants in front of her, she pays one euro to the Jury and moves forward in the queue by exactly $i$ positions. [*]If contestant $C_i$ has fewer than $i$ other contestants in front of her, the restaurant opens and process ends. [/list] [/list] [list=a] [*]Prove that the process cannot continue indefinitely, regardless of the Jury’s choices. [*]Determine for every $n$ the maximum number of euros that the Jury can collect by cunningly choosing the initial order and the sequence of moves. [/list]

Let us define a function $f:\mathbb N\to\mathbb N_0$ by $f(1)=0$ and, for all $n\in\mathbb N$, $$f(2n)=2f(n)+1,\qquad f(2n+1)=2f(n).$$Given a positive integer $p$, define a sequence $(u_n)$ by $u_0=p$ and $u_{k+1}=f(u_k)$ whenever $u_k\ne0$. (a) Prove that, for each $p\in\mathbb N$, there is a unique integer $v(p)$ such that $u_{v(p)}=0$. (b) Compute $v(1994)$. What is the smallest integer $p>0$ for which $v(p)=v(1994)$. (c) Given an integer $N$, determine the smallest integer $p$ such that $v(p)=N$.

For any real numbers sequence $\{x_n\}$ ,suppose that $\{y_n\}$ is a sequence such that: $y_1=x_1, y_{n+1}=x_{n+1}-(\sum\limits_{i = 1}^{n} {x^2_i})^{ \frac{1}{2}}$ ${(n \ge 1})$ . Find the smallest positive number $\lambda$ such that for any real numbers sequence $\{x_n\}$ and all positive integers $m$ , have $\frac{1}{m}\sum\limits_{i = 1}^{m} {x^2_i}\le\sum\limits_{i = 1}^{m} {\lambda^{m-i}y^2_i} .$ (High School Affiliated to Nanjing Normal University )

Find the greatest real number $ \alpha$ for which there exists a sequence of infinitive integers $ (a_n)$, ($ n \equal{} 1, 2, 3, \ldots$) satisfying the following conditions: 1) $ a_n > 1997n$ for every $ n \in\mathbb{N}^{*}$; 2) For every $ n\ge 2$, $ U_n\ge a^{\alpha}_n$, where $ U_n \equal{} \gcd\{a_i \plus{} a_k | i \plus{} k \equal{} n\}$.

The sequence $(L_n)$ is given by $L_0=2$, $L_1=1$, and $L_{n+1}=L_n+L_{n-1}$ for $n\ge1$. Prove that if a prime number $p$ divides $L_{2k}-2$ for $k\in\mathbb N$, then $p$ also divides $L_{2k+1}-1$.