Let $a_1,a_2,\dots,a_{2018}$ be a strictly increasing sequence of positive integers such that $$a_1+a_2+\cdots+a_{2018}=2018^{2018}.$$ What is the remainder when $a_1^3+a_2^3+\cdots+a_{2018}^3$ is divided by $6$? $\textbf{(A)}\ 0\qquad\textbf{(B)}\ 1\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}\ 3\qquad\textbf{(E)}\ 4$

Does there exist a sequence of positive integers $a_1,a_2,...$ such that every positive integer occurs exactly once and that the number $\tau (na_{n+1}^n+(n+1)a_n^{n+1})$ is divisible by $n$ for all positive integer. Here $\tau (n)$ denotes the number of positive divisor of $n$.

Random sequences $a_1, a_2, . . .$ and $b_1, b_2, . . .$ are chosen so that every element in each sequence is chosen independently and uniformly from the set $\{0, 1, 2, 3, . . . , 100\}$. Compute the expected value of the smallest nonnegative integer $s$ such that there exist positive integers $m$ and $n$ with $$s =\sum^m_{i=1} a_i =\sum^n_{j=1}b_j .$$

The Bernoulli sequence $\{B_{n}\}_{n \ge 0}$ is defined by \[B_{0}=1, \; B_{n}=-\frac{1}{n+1}\sum^{n}_{k=0}{{n+1}\choose k}B_{k}\;\; (n \ge 1)\] Show that for all $n \in \mathbb{N}$, \[(-1)^{n}B_{n}-\sum \frac{1}{p},\] is an integer where the summation is done over all primes $p$ such that $p| 2k-1$.

$(FRA 2)$ Let $n$ be an integer that is not divisible by any square greater than $1.$ Denote by $x_m$ the last digit of the number $x^m$ in the number system with base $n.$ For which integers $x$ is it possible for $x_m$ to be $0$? Prove that the sequence $x_m$ is periodic with period $t$ independent of $x.$ For which $x$ do we have $x_t = 1$. Prove that if $m$ and $x$ are relatively prime, then $0_m, 1_m, . . . , (n-1)_m$ are different numbers. Find the minimal period $t$ in terms of $n$. If n does not meet the given condition, prove that it is possible to have $x_m = 0 \neq x_1$ and that the sequence is periodic starting only from some number $k > 1.$

A high school basketball game between the Raiders and Wildcats was tied at the end of the first quarter. The number of points scored by the Raiders in each of the four quarters formed an increasing geometric sequence, and the number of points scored by the Wildcats in each of the four quarters formed an increasing arithmetic sequence. At the end of the fourth quarter, the Raiders had won by one point. Neither team scored more than $ 100$ points. What was the total number of points scored by the two teams in the first half? $ \textbf{(A)}\ 30 \qquad \textbf{(B)}\ 31 \qquad \textbf{(C)}\ 32 \qquad \textbf{(D)}\ 33 \qquad \textbf{(E)}\ 34$

Does there exist a sequence $(a_n)_{n\in\mathbb N}$ of distinct positive integers such that: (i) $a_n<1999n$ for all $n$; (ii) none of the $a_n$ contains three decimal digits $1$?

a) Let $\{a_n\}_{n=1}^\infty$ is sequence of integers bigger than 1. Proove that if $x>0$ is irrational, then $\ds x_n>\frac{1}{a_{n+1}}$ for infinitely many $n$, where $x_n$ is fractional part of $a_na_{n-1}\dots a_1x$. b)Find all sequences $\{a_n\}_{n=1}^\infty$ of positive integers, for which exist infinitely many $x\in(0,1)$ such that $\ds x_n>\frac{1}{a_{n+1}}$ for all $n$. [i]Nikolai Nikolov, Emil Kolev[/i]

For each positive integer $n$, let $a_n$ be the number of permutations $\tau$ of $\{1, 2, ... , n\}$ such that $\tau (\tau (\tau (x))) = x$ for $x = 1, 2, ..., n$. The first few values are $a_1 = 1, a_2 = 1, a_3 = 3, a_4 = 9$. Prove that $3^{334}$ divides $a_{2001}$. (A permutation of $\{1, 2, ... , n\}$ is a rearrangement of the numbers $\{1, 2, ... , n\}$ or equivalently, a one-to-one and onto function from $\{1, 2, ... , n\}$ to $\{1, 2, ... , n\}$. For example, one permutation of $\{1, 2, 3\}$ is the rearrangement $\{2, 1, 3\}$, which is equivalent to the function $\sigma : \{1, 2, 3\} \to \{1, 2, 3\}$ defined by $\sigma (1) = 2, \sigma (2) = 1, \sigma (3) = 3$.)

In the Cartesian plane, two integer points $(a_1, b_1$ and $(a_2, b_2)$ are connected if $(a_2, b_2)$ is one of the points $(-a_1, b_1 \pm 1)$, $(a_1 \pm 1,-b_1)$. Show that there exists an infinite sequence of integer points in which every integer point occurs, and every two consecutive points are connected.

The periods of two periodic sequences are $7$ and $13$. What is the maximal length of initial sections of the two sequences which can coincide? (The period $p$ of a sequence $a_1$,$a_2$, $...$ is the minimal $p$ such that $a_n = a_{n+p}$ for all $n$.) (AY Belov)

[url=http://www.mathlinks.ro/Forum/viewtopic.php?t=107262]IMO 2007 HKTST 1[/url] Problem 5 The sequence $\{a_{n}\}$ is defined by $a_{1}=0$ and $(n+1)^{3}a_{n+1}=2n^{2}(2n+1)a_{n}+2(3n+1)$ for all integers $\geq 1$. Show that infintely many members of the sequence are positive integers.

A sequence $x_1, x_2, ..., x_n, ...$ consists of an initial block of $p$ positive distinct integers that then repeat periodically. This means that $\{x_1, x_2, \dots, x_p\}$ are $p$ distinct positive integers and $x_{n+p}=x_n$ for every positive integer $n$. The terms of the sequence are not known and the goal is to find the period $p$. To do this, at each move it possible to reveal the value of a term of the sequence at your choice. (a) Knowing that $1 \le p \le 10$, find the least $n$ such that there is a strategy which allows to find $p$ revealing at most $n$ terms of the sequence. (b) Knowing that $p$ is one of the first $k$ prime numbers, find for which values of $k$ there exist a strategy that allows to find $p$ revealing at most $5$ terms of the sequence.

Let $(a_{n})_{n \ge 1}$ be a sequence of integers satisfying the inequality \[ 0\le a_{n-1}+\frac{1-\sqrt{5}}{2}a_{n}+a_{n+1} <1 \] for all $n \ge 2$. Prove that the sequence $(a_{n})$ is periodic. Any Hints or Sols for this hard problem?? :help:

In base $R_1$ the expanded fraction $F_1$ becomes $0.373737...$, and the expanded fraction $F_2$ becomes $0.737373...$. In base $R_2$ fraction $F_1$, when expanded, becomes $0.252525...$, while fraction $F_2$ becomes $0.525252...$. The sum of $R_1$ and $R_2$, each written in base ten is: $\text{(A)}\ 24 \qquad \text{(B)}\ 22\qquad \text{(C)}\ 21\qquad \text{(D)}\ 20\qquad \text{(E)}\ 19$