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?

Determine all non-negative integers $n$ for which there exist two relatively prime non-negative integers $x$ and $y$ and a positive integer $k\geqslant 2$ such that $3^n=x^k+y^k$.

Let $a_1, a_2, \ldots, a_{2024}$ be positive integers such that $a_{i+1}+1$ is a multiple of $a_i$ for all $i = 1, 2, \ldots , 2024$, with indices taken modulo $2024$. Find the maximum possible value of $a_1 + a_2 + \ldots + a_{2024}$. [i](Proposed by Ivan Chan Guan Yu)[/i]

Let $a$ and $b$ be two distinct positive integers with the same parity. Prove that the fraction $\frac{a!+b!}{2^a}$ is not an integer.

Find all odd primes $ p$, if any, such that $ p$ divides $ \sum_{n\equal{}1}^{103}n^{p\minus{}1}$

Let $n\ge 3$ be an integer and $a_1,a_2,\dots ,a_n$ be a finite sequence of positive integers, such that, for $k=2,3,\dots ,n$ $$n(a_k+1)-(n-1)a_{k-1}=1.$$ Prove that $a_n$ is not divisible by $(n-1)^2$.

Let $p$ be a prime number and $n$ a positive integer. Prove that the product \[{N=\frac{1}{p^{n^2}}} \prod_{i=1;2 \nmid i}^{2n-1} \biggl[ \left( (p-1)! \right) \binom{p^2 i}{pi}\biggr]\] Is a positive integer that is not divisible by $p.$

Let $a, b$ and $c$ be integers such that $a + b + c = 0$. (a) Show that $a^4 + b^4 + c^4$ is divisible by $a^2 + b^2 + c^2$. (b) Show that $a^{100} + b^{100} + c^{100}$ is divisible by $a^2 + b^2 + c^2$. .

Find all pairs $(x,y)$ of positive integers such that $x^{x+y} =y^{y-x}$.

A sequence of positive integers $a_1, a_2,\ldots, a_n$ is called [i]ladder[/i] of length $n$ if it consists of $n$ consecutive integers in ascending order. (a) Prove that for every positive integer $n$ there exist two ladders of length $n$, with no elements in common, $a_1, a_2,\ldots, a_n$ and $b_1, b_2,\ldots, b_n$, such that for all $i$ between $1$ and $n$, the greatest common divisor of $a_i$ and $b_i$ is equal to $1$. (b) Prove that for every positive integer $n$ there exist two ladders of length $n$, with no elements in common, $a_1, a_2,\ldots, a_n$ and $b_1, b_2,\ldots, b_n$, such that for all $i$ between $1$ and $n$, the greatest common divisor of $a_i$ and $b_i$ is greater than $1$.

Let $ p\ge 2 $ be a natural number that divides $ \binom{p}{k} , $ for any natural number $ k $ smaller than $ p. $ Prove that: [b]a)[/b] $ p $ is prime. [b]b)[/b] $ p^2 $ divides $ -2+\binom{2p}{p} . $

Find the largest integer $n$ such that $n$ is divisible by all positive integers less than $\sqrt[3]{n}$.

It is possible to partition the set $\{100, 101,\cdots , 1000\}$ into two subsets so that for any two distinct elements $x$ and $y$ belonging to the same subset $ \sqrt[3]{x + y}$ is irrational?

If $n$ is a positive integer, let $\phi(n)$ be the number of positive integers less than or equal to $n$ that are relatively prime to $n$. Compute the value of the infinite sum \[ \sum_{n=1}^\infty \frac{\phi(n) 2^n}{9^n - 2^n} \, . \]

Prove that for any prime ${p}$, different than ${2}$ and ${5}$, there exists such a multiple of ${p}$ whose digits are all nines. For example, if ${p = 13}$, such a multiple is ${999999 = 13 * 76923}$.

For every pair of positive integers $(n,m)$ with $n<m$, denote $s(n,m)$ be the number of positive integers such that the number is in the range $[n,m]$ and the number is coprime with $m$. Find all positive integers $m\ge 2$ such that $m$ satisfy these condition: i) $\frac{s(n,m)}{m-n} \ge \frac{s(1,m)}{m}$ for all $n=1,2,...,m-1$; ii) $2022^m+1$ is divisible by $m^2$

given a positive integer $n$. the set $\{ 1,2,..,2n \}$ is partitioned into $a_1<a_2<...<a_n $ and $b_1>b_2>...>b_n$. find the value of : $ \sum_{i=1}^{n}|a_i - b_i| $

Let $n \geq 2$ be a natural. Define $$X = \{ (a_1,a_2,\cdots,a_n) | a_k \in \{0,1,2,\cdots,k\}, k = 1,2,\cdots,n \}$$. For any two elements $s = (s_1,s_2,\cdots,s_n) \in X, t = (t_1,t_2,\cdots,t_n) \in X$, define $$s \vee t = (\max \{s_1,t_1\},\max \{s_2,t_2\}, \cdots , \max \{s_n,t_n\} )$$ $$s \wedge t = (\min \{s_1,t_1 \}, \min \{s_2,t_2,\}, \cdots, \min \{s_n,t_n\})$$ Find the largest possible size of a proper subset $A$ of $X$ such that for any $s,t \in A$, one has $s \vee t \in A, s \wedge t \in A$.

Find the largest constant $c>0$ such that for every positive integer $n\ge 2$, there always exist a positive divisor $d$ of $n$ such that $$d\le \sqrt{n}\hspace{0.5cm} \text{and} \hspace{0.5cm} \tau(d)\ge c\sqrt{\tau(n)}$$ where $\tau(n)$ is the number of divisors of $n$. [i]Proposed by Mohd. Suhaimi Ramly[/i]

Determine all pairs $(P, d)$ of a polynomial $P$ with integer coefficients and an integer $d$ such that the equation $P(x) - P(y) = d$ has infinitely many solutions in integers $x$ and $y$ with $x \neq y$.

Let $k$ be a given even positive integer. Sarah first picks a positive integer $N$ greater than $1$ and proceeds to alter it as follows: every minute, she chooses a prime divisor $p$ of the current value of $N$, and multiplies the current $N$ by $p^k -p^{-1}$ to produce the next value of $N$. Prove that there are infinitely many even positive integers $k$ such that, no matter what choices Sarah makes, her number $N$ will at some point be divisible by $2018$.

Let $\operatorname{cif}(x)$ denote the sum of the digits of the number $x$ in the decimal system. Put $a_1=1997^{1996^{1997}}$, and $a_{n+1}=\operatorname{cif}(a_n)$ for every $n>0$. Find $\lim_{n\to\infty}a_n$.

The prime number $p$ and a positive integer $k$ are given. Assume that $P(x)\in \mathbb Z[X]$ is a polynomial with coefficients in the set $\{0,1,\cdots,p-1\}$ with least degree which satisfies the following property: There exists a permutaion of numbers $1,2,\cdots,p-1$ around a circle such that for any $k$ consecutive numbers $a_1,a_2,\cdots,a_k$ one has $$ p | P(a_1)+P(a_2)+\cdots+ P(a_k). $$ Prove that $P(x)$ is of the form $ax^d+b$. Proposed by [i]Yahya Motevassel[/i]

Prove that there exist infinitely many positive integers which can't be represented as sum of less than $10$ odd positive integers' perfect squares.

A positive integer $N$ has the digits $1, 2, 3, 4, 5, 6$ and $7$, so that each digit $i$, $i \in \{1, 2, 3, 4, 5, 6, 7\}$ occurs $4i$ times in the decimal representation of $N$. Prove that $N$ is not a perfect square.