The sequence of integers $\{a_i\}_{i = 0}^{\infty}$ satisfies $a_0 = 3$, $a_1 = 4$, and \[a_{n+2} = a_{n+1} a_n + \left\lceil \sqrt{a_{n+1}^2 - 1} \sqrt{a_n^2 - 1}\right\rceil\] for $n \ge 0$. Evaluate the sum \[\sum_{n = 0}^{\infty} \left(\frac{a_{n+3}}{a_{n+2}} - \frac{a_{n+2}}{a_n} + \frac{a_{n+1}}{a_{n+3}} - \frac{a_n}{a_{n+1}}\right).\]

In an infinite sequence $a_1, a_2, a_3, \cdots$, the number $a_1$ equals $1$, and each $a_n, n > 1$, is obtained from $a_{n-1}$ as follows: [list]- if the greatest odd divisor of $n$ has residue $1$ modulo $4$, then $a_n = a_{n-1} + 1,$ - and if this residue equals $3$, then $a_n = a_{n-1} - 1.$[/list] Prove that in this sequence [b](a) [/b] the number $1$ occurs infinitely many times; [b](b)[/b] each positive integer occurs infinitely many times. (The initial terms of this sequence are $1, 2, 1, 2, 3, 2, 1, 2, 3, 4, 3, \cdots$ )

The sequence $ (x_n)$ is defined as; $ x_1\equal{}a$, $ x_2\equal{}b$ and for all positive integer $ n$, $ x_{n\plus{}2}\equal{}2008x_{n\plus{}1}\minus{}x_n$. Prove that there are some positive integers $ a,b$ such that $ 1\plus{}2006x_{n\plus{}1}x_n$ is a perfect square for all positive integer $ n$.

Consider the sequence $(a_n)_{n\ge 1}$ defined by $a_n=n$ for $n\in \{1,2,3.4,5,6\}$, and for $n \ge 7$: $$a_n={\lfloor}\frac{a_1+a_2+...+a_{n-1}}{2}{\rfloor}$$ where ${\lfloor}x{\rfloor}$ is the greatest integer less than or equal to $x$. For example : ${\lfloor}2.4{\rfloor} = 2, {\lfloor}3{\rfloor} = 3$ and ${\lfloor}\pi {\rfloor}= 3$. For all integers $n \ge 2$, let $S_n = \{a_1,a_1,...,a_n\}- \{r_n\}$ where $r_n$ is the remainder when $a_1 + a_2 + ... + a_n$ is divided by $3$. The minus $-$ denotes the ''[i]remove it if it is there[/i]'' notation. For example : $S_4 = {2,3,4}$ because $r_4= 1$ so $1$ is removed from $\{1,2,3,4\}$. However $S_5= \{1,2,3,4,5\}$ betawe $r_5 = 0$ and $0$ is not in the set $\{1,2,3,4,5\}$. 1. Determine $S_7,S_8,S_9$ and $S_{10}$. 2. We say that a set $S_n$ for $n\ge 6$ is well-balanced if it can be partitioned into three pairwise disjoint subsets with equal sum. For example : $S_6 = \{1,2,3,4,5,6\} =\{1,6\}\cup \{2,5\}\cup \{3,4\}$ and $1 +6 = 2 + 5 = 3 + 4$. Prove that $S_7,S_8,S_9$ and $S_{10}$ are well-balanced . 3. Is the set $S_{2019}$ well-balanced? Justify your answer.

Suppose $\{a_n\}$ is a decreasing sequence of reals and $\lim\limits_{n\to\infty} a_n = 0$. If $S_{2^k} - 2^k a_{2^k} \leq 1$ for any positive integer $k$, show that $$\sum_{n=1}^{\infty} a_n \leq 1$$ (At here, $S_m = \sum_{n=1}^m a_n$ is a partial sum of $\{a_n\}$.)

We say that a function $f: \mathbb{Z}_{\ge 0} \times \mathbb{Z}_{\ge 0} \to \mathbb{Z}$ is [i]great[/i] if for any nonnegative integers $m$ and $n$, \[f(m + 1, n + 1) f(m, n) - f(m + 1, n) f(m, n + 1) = 1.\] If $A = (a_0, a_1, \dots)$ and $B = (b_0, b_1, \dots)$ are two sequences of integers, we write $A \sim B$ if there exists a great function $f$ satisfying $f(n, 0) = a_n$ and $f(0, n) = b_n$ for every nonnegative integer $n$ (in particular, $a_0 = b_0$). Prove that if $A$, $B$, $C$, and $D$ are four sequences of integers satisfying $A \sim B$, $B \sim C$, and $C \sim D$, then $D \sim A$. [i]Ankan Bhattacharya[/i]

Alice and Bob are playing the following game: They take turns writing on the board natural numbers not exceeding $2018$ (to write the number twice is forbidden). Alice begins. A player wins if after his or her move there appear three numbers on the board which are in arithmetic progression. Which player has a winning strategy?

[u]Set 1 [/u] [b]1.1[/b] Compute the number of real numbers x such that the sequence $x$, $x^2$, $x^3$,$ x^4$, $x^5$, $...$ eventually repeats. (To be clear, we say a sequence “eventually repeats” if there is some block of consecutive digits that repeats past some point—for instance, the sequence $1$, $2$, $3$, $4$, $5$, $6$, $5$, $6$, $5$, $6$, $...$ is eventually repeating with repeating block $5$, $6$.) [b]1.2[/b] Let $T$ be the answer to the previous problem. Nicole has a broken calculator which, when told to multiply $a$ by $b$, starts by multiplying $a$ by $b$, but then multiplies that product by b again, and then adds $b$ to the result. Nicole inputs the computation “$k \times k$” into the calculator for some real number $k$ and gets an answer of $10T$. If she instead used a working calculator, what answer should she have gotten? [b]1.3[/b] Let $T$ be the answer to the previous problem. Find the positive difference between the largest and smallest perfect squares that can be written as $x^2 + y^2$ for integers $x, y$ satisfying $\sqrt{T} \le x \le T$ and $\sqrt{T} \le y \le T$. PS. You should use hide for answers.

[b]a)[/b] Find the number of infinite sequences of integers $ \left( a_n \right)_{n\ge 1} $ that have the property that $ a_na_{n+2}a_{n+3}=-1, $ for any natural number $ n. $ [b]b)[/b] Prove that there is no infinite sequence of integers $ \left( b_n \right)_{n\ge 1} $ that have the property that $ b_nb_{n+2}b_{n+3}=2005, $ for any natural number $ n. $

A sequence of positive integer numbers $a_1,a_2,\ldots$ for $i \geq 3$ satisfies $$a_{i+1}=a_i+gcd(a_{i-1},a_{i-2})$$ Prove that there exist two positive integer numbers $N, M$, such that $a_{n+1}-a_n=M$ for all $n \geq N$

In how many different ways can you write $2016$ as the sum of a sequence of consecutive natural numbers?

We consider the real sequence $(x_n)$ defined by $x_0=0, x_1=1$ and $x_{n+2}=3x_{n+1}-2x_n$ for $n=0,1,...$ We define the sequence $(y_n)$ by $y_n=x_n^2+2^{n+2}$ for every non negative integer $n$. Prove that for every $n>0$, $y_n$ is the square of an odd integer

Compute the number of integers $1 \leq n \leq 1024$ such that the sequence $\lceil n \rceil$, $\lceil n/2 \rceil$, $\lceil n/4 \rceil$, $\lceil n/8 \rceil$, $\ldots$ does not contain any multiple of $5$. [i]Proposed by Sean Li[/i]

For every non-negative integer $i$, define the number $M(i)$ as follows: write $i$ down as a binary number, so that we have a string of zeroes and ones, if the number of ones in this string is even, then set $M(i) = 0$, otherwise set $M(i) = 1$. (The first terms of the sequence $M(i)$, $i = 0, 1, 2, ...$ are $0, 1, 1, 0, 1, 0, 0, 1,...$ ) (a) Consider the finite sequence $M(O), M(1), . . . , M(1000) $. Prove that there are at least $320$ terms in this sequence which are equal to their neighbour on the right : $M(i) = M(i + 1 )$ . (b) Consider the finite sequence $M(O), M(1), . . . , M(1000000)$ . Prove that the number of terms $M(i)$ such that $M(i) = M(i +7)$ is at least $450000$. (A Kanel)

Consider the function $f (x) = (x - F_1)(x - F_2) ...(x -F_{3030})$ with $(F_n)$ is the Fibonacci sequence, which defined as $F_1 = 1, F_2 = 2$, $F_{n+2 }=F_{n+1} + F_n$, $n \ge 1$. Suppose that on the range $(F_1, F_{3030})$, the function $|f (x)|$ takes on the maximum value at $x = x_0$. Prove that $x_0 > 2^{2018}$.