The integer numbers from $1$ to $2002$ are written in a blackboard in increasing order $1,2,\ldots, 2001,2002$. After that, somebody erases the numbers in the $ (3k+1)-th$ places i.e. $(1,4,7,\dots)$. After that, the same person erases the numbers in the $(3k+1)-th$ positions of the new list (in this case, $2,5,9,\ldots$). This process is repeated until one number remains. What is this number?

Positive integers $a, b, c$ have the property that: $\bullet$ $a$ gives remainder $2$ when divided by $b$, $\bullet$ $b$ gives remainder $2$ when divided by $c$, $\bullet$ $c$ gives remainder $4$ when divided by $a$. Prove that $c = 4$.

For every integer $d \geq 1$, let $M_d$ be the set of all positive integers that cannot be written as a sum of an arithmetic progression with difference $d$, having at least two terms and consisting of positive integers. Let $A = M_1$, $B = M_2 \setminus \{2 \}, C = M_3$. Prove that every $c \in C$ may be written in a unique way as $c = ab$ with $a \in A, b \in B.$

For all integral values of parameter $t$, find all integral solutions $(x,y)$ of the equation $$ y^2 = x^4-22x^3+43x^2+858x+t^2+10452(t+39)$$ .

Let $\{F_n\}_{n\in\mathbb{Z}}$ and $\{L_n\}_{n\in\mathbb{Z}}$ denote the Fibonacci and Lucas numbers, respectively, given by $$F_{n+1} = F_n + F_{n-1}\ \text{and}\ L_{n+1} = L_n + L_{n-1}\ \text{for all}\ n \geq 1$$with $F_0 = 0$, $F_1 = 1$, $L_0 = 2$, and $L_1 = 1$. Prove that for integers $n \geq 1$ and $j \geq 0$ [list=1] [*]$\sum \limits_{k=1}^n F_{k\pm j}L_{k\mp j}=F_{2n+1}-1+\begin{cases} 0, & \text{if}\ n\ \text{is even}\\ \left(-1\right)^{\pm j}F_{\pm 2j}, & \text{if}\ n\ \text{is odd} \end{cases}$ [*] $\sum \limits_{k=1}^nF_{k+j}F_{k-j}L_{k+j}L_{k-j}=\frac{F_{4n+2}-1-nL_{4j}}{5}$ [/list]

We are given two two-digit numbers , $x$ and $y$. It is known that $x$ is twice as big as $y$. One of the digits of $y$ is the sum, while the other digit of $y$ is the difference, of the digits of $x$ . Find the values of $x$ and $y$, proving that there are no others.

The number $0$ or $1$ is to be assigned to each of the $n$ vertices of a regular polygon. In how many different ways can this be done (if we consider two assignments that can be obtained one from the other through rotation in the plane of the polygon to be identical)?

Solve the equation in nonnegative integers $a,b,c$: $3^a+2^b+2015=3c!$ I.Gorodnin

How many positive integers $n$ between $1$ and $2024$ (both included) are there such that $\lfloor{\sqrt{n}}\rfloor$ divides $n$? (For $x \in \mathbb{R}, \lfloor{n}\rfloor$ denotes the greatest integer less than or equal to $x$.) $(A) 44$ $(B) 132$ $(C) 1012$ $(D) 2024$

Show that there exists an infinite arithmetic progression of natural numbers such that the first term is $16$ and the number of positive divisors of each term is divisible by $5$. Of all such sequences, find the one with the smallest possible common difference.

Given $f_1(x)=2x-2$ and, for $k\ge2$, defined $f_k(x)=f(f_{k-1}(x))$ to be a real-valued function of $x$. Find the remainder when $f_{2013}(2012)$ is divided by the prime $2011$.

Given that the base-$17$ integer $\overline{8323a02421_{17}}$ (where a is a base-$17$ digit) is divisible by $\overline{16_{10}}$, find $a$. Express your answer in base $10$. [i]Proposed by Jonathan Liu[/i]

Find, with proof, all positive integers $n$ for which $2^n + 12^n + 2011^n$ is a perfect square.

Prove that any integer greater than or equal to $7$ can be written as a sum of two relatively prime integers, both greater than 1. (Two integers are relatively prime if they share no common positive divisor other than $1$. For example, $22$ and 15 are relatively prime, and thus $37 = 22+15$ represents the number 37 in the desired way.)

Find all values of the parameter $a$ for which there are exactly $1998$ integers $x$ satisfying the inequality $$x^2 -\pi x +a < 0.$$

Does there exist an infinite set $\displaystyle{M}$ consisting of positive integers such that for any $\displaystyle{a,b \in M}$ with $\displaystyle{a < b}$ the sum $\displaystyle{a + b}$ is square-free? [b]Note.[/b] A positive integer is called square-free if no perfect square greater than $\displaystyle{1}$ divides it. [i]Proposed by Fedor Petrov, St. Petersburg State University.[/i]

Let $p \ge 2$ be a prime number and $\frac{a_p}{b_p}= 1 +\frac12+ .. +\frac{1}{p^2 -1}$, where $a_p$ and $b_p$ are two relatively prime positive integers. Compute gcd $(p, b_p)$.

Suppose $ \,G\,$ is a connected graph with $ \,k\,$ edges. Prove that it is possible to label the edges $ 1,2,\ldots ,k\,$ in such a way that at each vertex which belongs to two or more edges, the greatest common divisor of the integers labeling those edges is equal to 1. [b]Note: Graph-Definition[/b]. A [b]graph[/b] consists of a set of points, called vertices, together with a set of edges joining certain pairs of distinct vertices. Each pair of vertices $ \,u,v\,$ belongs to at most one edge. The graph $ G$ is connected if for each pair of distinct vertices $ \,x,y\,$ there is some sequence of vertices $ \,x \equal{} v_{0},v_{1},v_{2},\cdots ,v_{m} \equal{} y\,$ such that each pair $ \,v_{i},v_{i \plus{} 1}\;(0\leq i < m)\,$ is joined by an edge of $ \,G$.

The Blablabla set contains all the different seven-digit numbers that can be formed with the digits $2, 3, 4, 5, 6, 7$ and $8$. Prove that there are not two Blablabla numbers such that one of them is divisible by the other.

Let $a$ and $n$ be two positive integer numbers such that the (positive) prime factors of $a$ be all greater than $n$. Prove that $n!$ divides $(a - 1)(a^2 - 1)\cdots (a^{n-1} - 1)$. [i]AMM Magazine[/i]

A five-digit integer is said to be [i]balanced [/i]i f the sum of any three of its digits is divisible by any of the other two. How many [i]balanced [/i] numbers are there?

Let $p$ be an odd prime number and $M$ a set derived from $\frac{p^2 + 1}{2}$ square numbers. Investigate whether $p$ elements can be selected from this set whose arithmetic mean is an integer. (Walther Janous)

Given an integer $n\ge 2$, a function $f:\mathbb{Z}\rightarrow \{1,2,\ldots,n\}$ is called [i]good[/i], if for any integer $k,1\le k\le n-1$ there exists an integer $j(k)$ such that for every integer $m$ we have \[f(m+j(k))\equiv f(m+k)-f(m) \pmod{n+1}. \] Find the number of [i]good[/i] functions.

Compute the smallest positive integer $n$ for which there are at least two odd primes $p$ such that $\sum_{k=1}^{n} (-1)^{v_p(k!)} < 0$. Note: for a prime $p$ and a positive integer $m$, $v_p(m)$ is the exponent of the largest power of $p$ that divides $m$; for example, $v_3(18) = 2$.

There are $n$ coins lying in a circle. Each coin has two sides, $+$ and $-$. A $flop$ means to flip every coin that has two different neighbors simultaneously, while leaving the others alone. For instance, $++-+$, after one $flop$, becomes $+---$. For $n$ coins, let us define $M$ to be a $perfect$ $number$ if for any initial arrangement of the coins, the arrangement of the coins after $m$ $flops$ is exactly the same as the initial one. (a) When $n=1024$, find a perfect number $M$. (b) Find all $n$ for which a perfect number $M$ exist.