Let $m$ be a fixed integer greater than $1$. The sequence $x_0$, $x_1$, $x_2$, $\ldots$ is defined as follows: \[x_i = \begin{cases}2^i&\text{if }0\leq i \leq m - 1;\\\sum_{j=1}^mx_{i-j}&\text{if }i\geq m.\end{cases}\] Find the greatest $k$ for which the sequence contains $k$ consecutive terms divisible by $m$ . [i]Proposed by Marcin Kuczma, Poland[/i]

We define an operation $\oplus$ on the set $\{0, 1\}$ by \[ 0 \oplus 0 = 0 \,, 0 \oplus 1 = 1 \,, 1 \oplus 0 = 1 \,, 1 \oplus 1 = 0 \,.\] For two natural numbers $a$ and $b$, which are written in base $2$ as $a = (a_1a_2 \ldots a_k)_2$ and $b = (b_1b_2 \ldots b_k)_2$ (possibly with leading 0's), we define $a \oplus b = c$ where $c$ written in base $2$ is $(c_1c_2 \ldots c_k)_2$ with $c_i = a_i \oplus b_i$, for $1 \le i \le k$. For example, we have $7 \oplus 3 = 4$ since $ 7 = (111)_2$ and $3 = (011)_2$. For a natural number $n$, let $f(n) = n \oplus \left[ n/2 \right]$, where $\left[ x \right]$ denotes the largest integer less than or equal to $x$. Prove that $f$ is a bijection on the set of natural numbers.

Let $p = 10009$ be a prime number. Determine the number of ordered pairs of integers $(x,y)$ such that $1\le x,y \le p$ and $x^3-3xy+y^3+1$ is divisible by $p$.

If $x$ and $y$ are real numbers, prove that $\frac{4x^2+1}{y^2+2}$ is not integer

Find all pairs $(n, q)$ such that $n$ is a positive integer, $q$ is a not integer rational and $$n^q-q$$ is an integer.

Find all the integers $x$ in $[20, 50]$ such that $6x+5\equiv 19 \mod 10$, that is, $10$ divides $(6x+15)+19$.

Determine if there are positive integers $a, b$ such that all terms of the sequence defined by \[ x_{1}= 2010,x_{2}= 2011\\ x_{n+2}= x_{n}+ x_{n+1}+a\sqrt{x_{n}x_{n+1}+b}\quad (n\ge 1) \] are integers.

In an infinite increasing sequence of positive integers, every term from the $2002^{\text{th}}$ term divides the sum of all preceding terms. Prove that every term starting from some term is equal to the sum of all preceding terms.

Consider the set $I=\{ 1,2, \cdots, 2020 \}$. Let $W= \{w(a,b)=(a+b)+ab | a,b \in I \} \cap I$, $Y=\{y(a,b)=(a+b) \cdot ab | a,b \in I \} \cap I$ be its two subsets. Further, let $X= W \cap Y$. [b](1)[/b] Find the sum of maximal and minimal elements in $X$. [b](2)[/b] An element $n=y(a,b) (a \le b)$ in $Y$ is called [i]excellent[/i], if its representation is not unique (for instance, $20=y(1,5)=y(2,3)$). Find the number of [i]excellent[/i] elements in $Y$. [hide=Note][b](2)[/b] is only for Grade 11.[/hide]

Determine all $x$, $y$, $k$ and $n$ positive integers such that: $10^x$ + $10^y$ + $n!$ = $2024^k$

Define $S_r(n)$: digit sum of $n$ in base $r$. For example, $38=(1102)_3,S_3(38)=1+1+0+2=4$. Prove: [b](a)[/b] For any $r>2$, there exists prime $p$, for any positive intenger $n$, $S_{r}(n)\equiv n\mod p$. [b](b)[/b] For any $r>1$ and prime $p$, there exists infinitely many $n$, $S_{r}(n)\equiv n\mod p$.

Find $ a,b,c \in N$ such that $ ab$ divides $ a^2\plus{}b^2\plus{}1$.

Given a finite sequence with terms in the set $A=\{0,1,…,121\}$ , it is allowed to replace each term by a number from the set $A$ so that like terms are replaced by like numbers, and different terms by different numbers. (Terms may remain without replacement.) The objective is to obtain, from a given sequence, through several such changes, a new sequence with sum divisible by $121$ . Show that it is possible to achieve the objective for every initial sequence. [hide=original wording]Dada una secuencia finita con términos en el conjunto A={0,1,…,121} , está permitido reemplazar cada término por un número del conjunto A de modo que términos iguales se reemplacen por números iguales, y términos distintos por números distintos. (Pueden quedar términos sin reemplazar.) El objetivo es obtener, a partir de una sucesión dada, mediante varios de tales cambios, una nueva sucesión con suma divisible por 121 . Demostrar que es posible lograr el objetivo para toda sucesión inicial.[/hide]

Find all polynomials $P(x)$ of odd degree $d$ and with integer coefficients satisfying the following property: for each positive integer $n$, there exists $n$ positive integers $x_1, x_2, \ldots, x_n$ such that $\frac12 < \frac{P(x_i)}{P(x_j)} < 2$ and $\frac{P(x_i)}{P(x_j)}$ is the $d$-th power of a rational number for every pair of indices $i$ and $j$ with $1 \leq i, j \leq n$.

Let $n \geqslant 100$ be an integer. Ivan writes the numbers $n, n+1, \ldots, 2 n$ each on different cards. He then shuffles these $n+1$ cards, and divides them into two piles. Prove that at least one of the piles contains two cards such that the sum of their numbers is a perfect square.

Let $a > 1$ be a positive integer. Prove that for some nonnegative integer $n$, the number $2^{2^n}+a$ is not prime. [i]Proposed by Jack Gurev[/i]

Find all functions $f : \mathbb{N} \rightarrow \mathbb{N}$ for which these two conditions hold simultaneously (i) For all $m,n \in \mathbb{N}$ we have: $$ \frac{f(mn)}{\gcd(m,n)} = \frac{f(m)f(n)}{f(\gcd(m,n))};$$ (ii) For all prime numbers $p$, there exists a prime number $q$ such that $f(p^{2025})=q^{2025}$.

[b]p7.[/b] Let $A, B, C$, and $D$ be equally spaced points on a circle $O$. $13$ circles of equal radius lie inside $O$ in the configuration below, where all centers lie on $\overline{AC}$ or $\overline{BD}$, adjacent circles are externally tangent, and the outer circles are internally tangent to $O$. Find the ratio of the area of the region inside $O$ but outside the smaller circles to the total area of the smaller circles. [img]https://cdn.artofproblemsolving.com/attachments/9/7/7ff192baf58f40df0e4cfae4009836eab57094.png[/img] [b]p8.[/b] Find the greatest divisor of $40!$ that has exactly three divisors. [b]p9.[/b] Suppose we have positive integers $a, b, c$ such that $a = 30$, lcm $(a, b) = 210$, lcm $(b, c) = 126$. What is the minimum value of lcm $(a, c)$? PS. You should use hide for answers.

$n$ consecutive positive integers are put down in a row (not necessarily in order) so that the sum of any three successive integers in the row is divisible by the leftmost number in the triple. What is the largest possible value of $n$ if the last number in the row is odd? (A Shapovalov)

$P(x)$ and $Q(x)$ are two polynomials with integer coefficients such that $P(x)|Q(x)^2+1$. [b]a)[/b] Prove that there exists polynomials $A(x)$ and $B(x)$ with rational coefficients and a rational number $c$ such that $P(x)=c(A(x)^2+B(x)^2)$. [b]b)[/b] If $P(x)$ is a monic polynomial with integer coefficients, Prove that there exists two polynomials $A(x)$ and $B(x)$ with integer coefficients such that $P(x)$ can be written in the form of $A(x)^2+B(x)^2$. [i]Proposed by Mohammad Gharakhani[/i]

Find all positive integers $n=p_1p_2 \cdots p_k$ which divide $(p_1+1)(p_2+1)\cdots (p_k+1)$ where $p_1 p_2 \cdots p_k$ is the factorization of $n$ into prime factors (not necessarily all distinct).

Let $\, k_1 < k_2 < k_3 < \cdots \,$ be positive integers, no two consecutive, and let $\, s_m = k_1 + k_2 + \cdots + k_m \,$ for $\, m = 1,2,3, \ldots \; \;$. Prove that, for each positive integer $\, n, \,$ the interval $\, [s_n, s_{n+1}) \,$ contains at least one perfect square.

For which $n{}$ is it possible that a product of $n{}$ consecutive positive integers is equal to a sum of $n{}$ consecutive (not necessarily the same) positive integers? [i]Boris Frenkin[/i]

Define the modulo $2553$ distance $d(x, y)$ between two integers $x, y$ to be the smallest nonnegative integer $d$ equivalent to either $x - y$ or $y - x$ modulo $2553$. Show that, given a set S of integers such that $|S| \ge 70$, there must be $m, n \in S$ with $d(m, n) \le 36$.

Let $n$ be a positive integer. Prove that the number of lines which go through the origin and precisely one other point with integer coordinates $(x,y),0\le x,y\le n$, is at least $\frac{n^2}{4}$.