A finite arithmetic progression of positive integers $a_1,a_2,\ldots,a_n$ satisfies the condition that for all $1\leq i<j\leq n$, the number of positive divisors of $\gcd\left(a_i,a_j\right)$ is equal to $j-i$. Find the maximum possible value of $n$. $\text{(A) }2\qquad\text{(B) }3\qquad\text{(C) }4\qquad\text{(D) }5\qquad\text{(E) }6$

Prove that you can pick $15$ distinct positive integers between $1$ and $2023$, such that each one of them and the sum between some of them is never a perfect square, nor a perfect cube or any other greater perfect power.

Let $a, b$, and $c$ be integers that satisfy $2a + 3b = 52$, $3b + c = 41$, and $bc = 60$. Find $a + b + c$

Does there exist positive integer $n$ such that sum of all digits of number $n(4n+1)$ is equal to $2017$

$n>1$ is odd number. There are numbers $n,n+1,n+2,...,2n-1$ on the blackboard. Prove that we can erase one number, such that the sum of all numbers will be not divided any number on the blackboard.

The equation $166\times 56 = 8590$ is valid in some base $b \ge 10$ (that is, $1, 6, 5, 8, 9, 0$ are digits in base $b$ in the above equation). Find the sum of all possible values of $b \ge 10$ satisfying the equation.

Prove that $x^3+y^3+z^3+t^3=1999$ has infinitely many soln. over $\mathbb{Z}$.

Show that there exists a positive real $C$ such that for any naturals $H,N$ satisfying $H \geq 3, N \geq e^{CH}$, for any subset of $\{1,2,\ldots,N\}$ with size $\lceil \frac{CHN}{\ln N} \rceil$, one can find $H$ naturals in it such that the greatest common divisor of any two elements is the greatest common divisor of all $H$ elements.

For each positive integer $n$, let $rad(n)$ denote the product of the distinct prime factors of $n$. Show that there exists integers $a,b > 1$ such that $gcd(a,b)=1$ and $$rad(ab(a+b)) < \frac{a+b}{2024^{2024}}$$. For example, $rad(20)=rad(2^2\cdot 5)=2\cdot 5=10$.

Positive rational number $a$ and $b$ satisfy the equality \[a^3 + 4a^2b = 4a^2 + b^4.\] Prove that the number $\sqrt{a}-1$ is a square of a rational number.

We have a positive integer $ n$ such that $ n \neq 3k$. Prove that there exists a positive integer $ m$ such that $ \forall_{k\in N \ k\geq m} \ k$ can be represented as a sum of digits of some multiplication of $ n$.

Determine the least positive integer $n$ with the following property – for every 3-coloring of numbers $1,2,\ldots,n$ there are two (different) numbers $a,b$ of the same color such that $|a-b|$ is a perfect square.

Let $P(x)$ be a polynomial with integer coefficients such that $P(0)=1$, and let $c > 1$ be an integer. Define $x_0=0$ and $x_{i+1} = P(x_i)$ for all integers $i \ge 0$. Show that there are infinitely many positive integers $n$ such that $\gcd (x_n, n+c)=1$. [i]Proposed by Milan Haiman and Carl Schildkraut[/i]

Let $f : \{ 1, 2, 3, \dots \} \to \{ 2, 3, \dots \}$ be a function such that $f(m + n) | f(m) + f(n) $ for all pairs $m,n$ of positive integers. Prove that there exists a positive integer $c > 1$ which divides all values of $f$.

Let \( n \geq 4 \). Proof that \[ (2^x - 1)(5^x - 1) = y^n \] have no positive integer solution \((x, y)\).

Let $ p \geq 2$ be a prime number. Eduardo and Fernando play the following game making moves alternately: in each move, the current player chooses an index $i$ in the set $\{0,1,2,\ldots, p-1 \}$ that was not chosen before by either of the two players and then chooses an element $a_i$ from the set $\{0,1,2,3,4,5,6,7,8,9\}$. Eduardo has the first move. The game ends after all the indices have been chosen .Then the following number is computed: $$M=a_0+a_110+a_210^2+\cdots+a_{p-1}10^{p-1}= \sum_{i=0}^{p-1}a_i.10^i$$. The goal of Eduardo is to make $M$ divisible by $p$, and the goal of Fernando is to prevent this. Prove that Eduardo has a winning strategy. [i]Proposed by Amine Natik, Morocco[/i]

Prove that $n^2 + 3n + 5$ is not divisible by $121$ for any positive integer $n$.

In numbers $1010... 101$ Ones and zeros alternate, if there are $n$ ones, there are $n -1$ zeros ($n \ge 2$ ).Determine the values of $n$ for which the number $1010... 101$, which has $n$ ones, is prime.

Find the minimum of the expression $\frac{a + b + c}{2} -\frac{[a, b] + [b, c] + [c, a]}{a + b + c}$ where the variables $a, b, c$ are any integers greater than $1$ and $[x, y]$ denotes the least common multiple of numbers $x, y$.

Find all pairs of integers $a$ and $b$ for which \[7a+14b=5a^2+5ab+5b^2\]

Given are the integers $a , b , c, \alpha, \beta$ and the sequence $(u_n)$ is defined by $u_1=\alpha, u_2=\beta, u_{n+2}=au_{n+1}+bu_n+c$ for all $n \geq 1$. a) Prove that if $a = 3 , b= -2 , c = -1$ then there are infinitely many pairs of integers $(\alpha ; \beta)$ so that $u_{2023}=2^{2022}$. b) Prove that there exists a positive integer $n_0$ such that only one of the following two statements is true: i) There are infinitely many positive integers $m$, such that $u_{n_0}u_{n_0+1}\ldots u_{n_0+m}$ is divisible by $7^{2023}$ or $17^{2023}$ ii) There are infinitely many positive integers $k$ so that $u_{n_0}u_{n_0+1}\ldots u_{n_0+k}-1$ is divisible by $2023$

Find all pairs $(p,q)$ of positive primes such that the equation $3x^2 - px + q = 0$ has two distinct rational roots.

Let $a$ and $b$ be two positive integers. Prove that the integer \[a^2+\left\lceil\frac{4a^2}b\right\rceil\] is not a square. (Here $\lceil z\rceil$ denotes the least integer greater than or equal to $z$.) [i]Russia[/i]

Let $p > 10^9$ be a prime number such that $4p + 1$ is also prime. Prove that the decimal expansion of $\frac{1}{4p+1}$ contains all the digits $0,1, \ldots, 9$.

Let $n\ge 2$ be an integer. $a_1,a_2,\dotsc,a_n$ are arbitrarily chosen positive integers with $(a_1,a_2,\dotsc,a_n)=1$. Let $A=a_1+a_2+\dotsb+a_n$ and $(A,a_i)=d_i$. Let $(a_2,a_3,\dotsc,a_n)=D_1, (a_1,a_3,\dotsc,a_n)=D_2,\dotsc, (a_1,a_2,\dotsc,a_{n-1})=D_n$. Find the minimum of $\prod\limits_{i=1}^n\dfrac{A-a_i}{d_iD_i}$