Find all positive integers $n$ with the property that the set \[\{n,n+1,n+2,n+3,n+4,n+5\}\] can be partitioned into two sets such that the product of the numbers in one set equals the product of the numbers in the other set.

Let $a, b, c, n$ be four integers, where n$\ge 2$, and let $p$ be a prime dividing both $a^2+ab+b^2$ and $a^n+b^n+c^n$, but not $a+b+c$. for instance, $a \equiv b \equiv -1 (mod \,\, 3), c \equiv 1 (mod \,\, 3), n$ a positive even integer, and $p = 3$ or $a = 4, b = 7, c = -13, n = 5$, and $p = 31$ satisfy these conditions. Show that $n$ and $p - 1$ are not coprime.

Let $N_0$ be the set of all non-negative integers. Let $f : N_0 \times N_0 \to N_0$ be a function such that for all non-negative integers $a,b$: $$f(a,b) = f(b,a),$$ $$f(a,0) = 0,$$ $$f(a+b,b) = f(a,b) +b.$$ Compute $$\sum_{i=0}^{30}\sum_{j=0}^{2^i-1}f(2^i, j)$$

Let $d(n)$ be the quantity of positive divisors of $n$, for example $d(1)=1,d(2)=2,d(10)=4$. The [b]size[/b] of $n$ is $k$ if $k$ is the least positive integer, such that $d^k(n)=2$. Note that $d^s(n)=d(d^{s-1}(n))$. a) How many numbers in the interval $[3,1000]$ have size $2$ ? b) Determine the greatest size of a number in the interval $[3,1000]$.

A positive integer $n$ is $inverosimil$ if there exists $n$ integers not necessarily distinct such that the sum and the product of this integers are equal to $n$. How many positive integers less than or equal to $2022$ are $inverosimils$?

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]

Define $a_k = (k^2 + 1)k!$ and $b_k = a_1 + a_2 + a_3 + \cdots + a_k$. Let \[\frac{a_{100}}{b_{100}} = \frac{m}{n}\] where $m$ and $n$ are relatively prime natural numbers. Find $n - m$.

For any integer $n \ge 2$, let $b_n$ be the least positive integer such that, for any integer $N$, $m$ divides $N$ whenever $m$ divides the digit sum of $N$ written in base $b_n$, for $2 \le m \le n$. Find the integer nearest to $b_{36}/b_{25}$.

Let $f(x) = 3x + 2.$ Prove that there exists $m \in \mathbb{N}$ such that $f^{100}(m)$ is divisible by $1988$.

$f(n+f(n))=f(n)$ for every $n \in \mathbb{N}$. a)Prove, that if $f(n)$ is finite, then $f$ is periodic. b) Give example nonperiodic function. PS. $0 \not \in \mathbb{N}$

Let $a_1,a_2,\dots, a_n$ be positive integers with product $P,$ where $n$ is an odd positive integer. Prove that $$\gcd(a_1^n+P,a_2^n+P,\dots, a_n^n+P)\le 2\gcd(a_1,\dots, a_n)^n.$$ [i]Proposed by Daniel Liu[/i]

Let $p, q$ be two distinct primes. Prove that there are positive integers $a, b$ such that the arithmetic mean of all positive divisors of the number $n = p^aq^b$ is an integer.

Let $a$, $b$, $c$ and $d$ be integers such that $ac$, $bd$ and $bc+ad$ are divisible with positive integer $m$. Show that numbers $bc$ and $ad$ are divisible with $m$

A positive integer $t$ is called a Jane's integer if $t = x^3+y^2$ for some positive integers $x$ and $y$. Prove that for every integer $n \ge 2$ there exist infinitely many positive integers $m$ such that the set of $n^2$ consecutive integers $\{m+1,m+2,\dotsc,m+n^2\}$ contains exactly $n + 1$ Jane's integers.

Find all pairs $(n,d)$ of positive integers such that $d\mid n^2$ and $(n-d)^2<2d$. [i]Linus Tang[/i]

Prove that there are no integers $x$ and $y$ for which $$x^2 +3xy-2y^2 =122.$$

Determine all triples of positive integers $a, b, c$ that satisfy a) $[a, b] + [a, c] + [b, c] = [a, b, c]$. b) $[a, b] + [a, c] + [b, c] = [a, b, c] + (a, b, c)$. Remark: Here $[x, y$] denotes the least common multiple of positive integers $x$ and $y$, and $(x, y)$ denotes their greatest common divisor.

Let $n$ and $k$ be relatively prime positive integers with $k<n$. Each number in the set $M=\{1,2,3,\ldots,n-1\}$ is colored either blue or white. For each $i$ in $M$, both $i$ and $n-i$ have the same color. For each $i\ne k$ in $M$ both $i$ and $|i-k|$ have the same color. Prove that all numbers in $M$ must have the same color.

Let \( x \), \( y \), \( p \) be positive integers that satisfy the equation \( x^4 = p + 9y^4 \), where \( p \) is a prime number. Show that \( \frac{p^2 - 1}{3} \) is a perfect square and a multiple of 16.

Find the remainder when $$20^{20}+21^{21}-21^{20}-20^{21}$$ is divided by $100$.

Find the largest positive integer $n$ such that for each prime $p$ with $2<p<n$ the difference $n-p$ is also prime.

The number $1$ is written on the blackboard. At any point, Kornny may pick two (not necessary distinct) of the numbers $a$ and $b$ written on the board and write either $ab$ or $\frac{1}{a}+\frac{1}{b}+\frac{1}{ab}$ on the board as well. Determine all possible numbers that Kornny can write on the board in finitely many steps.

For positive integers $i$ and $j$, define $d(i,j)$ as follows: $d(1,j) = 1, d(i,1) = 1$ for all $i$ and $j$, and for $i, j > 1$, $d(i,j) = d(i-1,j) + d(i,j-1) + d(i-1,j-1)$. Compute the remainder when $d(3,2016)$ is divided by $1000$.

Find all integers $n$ such that $n^{n-1}-1$ is square-free.

Consider $(f_n)_{n\ge 0}$ the Fibonacci sequence, defined by $f_0 = 0$, $f_1 = 1$, $f_{n+1} = f_n + f_{n-1}$ for all positive integers $n$. Solve the following equation in positive integers $$nf_nf_{n+1} = (f_{n+2} - 1)^2.$$ .