Let $ \left(a_n \right)_{n \in \mathbb{N}}$ defined by $ a_1 \equal{} 1,$ and $ a_{n \plus{} 1} \equal{} a^4_n \minus{} a^3_n \plus{} 2a^2_n \plus{} 1$ for $ n \geq 1.$ Show that there is an infinite number of primes $ p$ such that none of the $ a_n$ is divisible by $ p.$

Given a sequence of natural numbers $ (a_i) $, for each natural number $ n $ the sum of the terms of the sequence that are not greater than $ n $ is a number not less than $ n $. Prove that for every natural number $ k $ it is possible to choose from the sequence $ (a_i) $ a finite sequence with the sum of terms equal to $ k $.

For any positive integer $a,$ $\sigma(a)$ denotes the sum of the positive integer divisors of $a.$ Let $n$ be the least positive integer such that $\sigma(a^n)-1$ is divisible by $2021$ for all positive integers $a.$ Find the sum of the prime factors in the prime factorization of $n.$

How many zeros does $101^{100} - 1$ end with?

Determine all integers $s \ge 4$ for which there exist positive integers $a$, $b$, $c$, $d$ such that $s = a+b+c+d$ and $s$ divides $abc+abd+acd+bcd$. [i]Proposed by Ankan Bhattacharya and Michael Ren[/i]

Find all prime numbers $ p,q$ less than 2005 and such that $ q|p^2 \plus{} 4$, $ p|q^2 \plus{} 4$.

Find all primes $p \ge 3$ such that $p- \lfloor p/q \rfloor q$ is a square-free integer for any prime $q<p$.

If permutations of the numbers $2, 3,4,..., 102$ are denoted by $a_i,a_2, a_3,...,a_{101}$, find all such permutations in which $a_k$ is divisible by $k$ for all $k$.

For integer $n\geq4$, find the smallest integer $f(n)$, such that for any positive integer $m$, in any subset with $f(n)$ elements of the set $\{m, m+1, \cdots, m+n-1\}$ there are at least three elements that are relatively prime .

Consider a finite set of points, $\Phi$, in the $\mathbb{R}^2$, such that they follow the following properties : [list] [*] $\Phi$ doesn't contain the origin $\{(0,0)\}$ and not all points are collinear. [*] If $\alpha \in \Phi$, then $-\alpha \in \Phi$, $c\alpha \notin \Phi $ for $c\neq 1$ or $-1$ [*] If $\alpha, \ \beta$ are in $\Phi$, then the reflection of $\beta$ in the line passing through the origin and perpendicular to the line containing origin and $\alpha$ is in $\Phi$ [*] If $\alpha = (a,b) , \ \beta = (c,d)$, (both $\alpha, \ \beta \in \Phi$) then $\frac{2(ac+bd)}{c^2+d^2} \in \mathbb{Z}$ [/list] Prove that there cannot be 5 collinear points in $\Phi$

Find all positive integers $n$ for which the number $1!+2!+3!+\cdots+n!$ is a perfect power of an integer.

Dave rolls a fair six-sided die until a six appears for the first time. Independently, Linda rolls a fair six-sided die until a six appears for the first time. Let $ m$ and $ n$ be relatively prime positive integers such that $ \frac{m}{n}$ is the probability that the number of times Dave rolls his die is equal to or within one of the number of times Linda rolls her die. Find $ m\plus{}n$.

Find all positive integers $n$, such that there exists a positive integer $m$ and primes $1<p<q$ such that $q-p \mid m$ and $p, q \mid n^m+1$.

Let the sequence $a_1,a_2,\ldots,a_n,\ldots$ is defined by the conditions: $a_1=2$ and $a_{n+1}=a_n^2-a_n+1$ $(n=1,2,\ldots)$. Prove that: (a) $a_m$ and $a_n$ are relatively prime numbers when $m\ne n$. (b) $\lim_{n\to\infty}\sum_{k=1}^n\frac1{a_k}=1$ [i]I. Tonov[/i]

Consider a triple $(a, b, c)$ of pairwise distinct positive integers satisfying $a + b + c = 2013$. A step consists of replacing the triple $(x, y, z)$ by the triple $(y + z - x,z + x - y,x + y - z)$. Prove that, starting from the given triple $(a, b,c)$, after $10$ steps we obtain a triple containing at least one negative number.

[b]p1.[/b] A boy has as many sisters as brothers. How ever, his sister has twice as many brothers as sisters. How many boys and girls are there in the family? [b]p2.[/b] Solve each of the following problems. (1) Find a pair of numbers with a sum of $11$ and a product of $24$. (2) Find a pair of numbers with a sum of $40$ and a product of $400$. (3) Find three consecutive numbers with a sum of $333$. (4) Find two consecutive numbers with a product of $182$. [b]p3.[/b] $2008$ integers are written on a piece of paper. It is known that the sum of any $100$ numbers is positive. Show that the sum of all numbers is positive. [b]p4.[/b] Let $p$ and $q$ be prime numbers greater than $3$. Prove that $p^2 - q^2$ is divisible by $24$. [b]p5.[/b] Four villages $A,B,C$, and $D$ are connected by trails as shown on the map. [img]https://cdn.artofproblemsolving.com/attachments/4/9/33ecc416792dacba65930caa61adbae09b8296.png[/img] On each path $A \to B \to C$ and $B \to C \to D$ there are $10$ hills, on the path $A \to B \to D$ there are $22$ hills, on the path $A \to D \to B$ there are $45$ hills. A group of tourists starts from $A$ and wants to reach $D$. They choose the path with the minimal number of hills. What is the best path for them? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all positive integers $n$ such that there exists a sequence of positive integers $a_1$, $a_2$,$\ldots$, $a_n$ satisfying: \[a_{k+1}=\frac{a_k^2+1}{a_{k-1}+1}-1\] for every $k$ with $2\leq k\leq n-1$. [i]Proposed by North Korea[/i]

Denote by $\mathbb{N}$ the set of all positive integers. Find all functions $f:\mathbb{N}\rightarrow \mathbb{N}$ such that for all positive integers $m$ and $n$, the integer $f(m)+f(n)-mn$ is nonzero and divides $mf(m)+nf(n)$. [i]Proposed by Dorlir Ahmeti, Albania[/i]

(a) Prove that in the set $ S=\{2008,2009,. . .,4200\}$ there are $ 5^3$ elements such that any three of them are not in arithmetic progression. (b) Bonus: Try to find a smaller integer $ n \in (2008,4200)$ such that in the set $ S'=\{2008,2009,...,n\}$ there are $ 5^3$ elements such that any three of them are not in arithmetic progression.

Let $a_1, \dots, a_n, b_1, \dots, b_n$ be $2n$ positive integers such that the $n+1$ products \[a_1 a_2 a_3 \cdots a_n, b_1 a_2 a_3 \cdots a_n, b_1 b_2 a_3 \cdots a_n, \dots, b_1 b_2 b_3 \cdots b_n\] form a strictly increasing arithmetic progression in that order. Determine the smallest possible integer that could be the common difference of such an arithmetic progression.

Find all natural numbers $n$ and nonnegative integers $x_{1},x_{2},...,x_{n}$ such that $\sum_{i=1}^{n}x_{i}^{2}=1+\frac{4}{4n+1}(\sum_{i=1}^{n}x_{i})^{2}$.

positive reals $a_1, a_2, . . . $ satisfying (i) $a_{n+1}=a_1^2\cdot a_2^2 \cdot . . . \cdot a_n^2-3$(all positive integers $n$) (ii) $\frac{1}{2}(a_1+\sqrt{a_2-1})$ is positive integer. prove that $\frac{1}{2}(a_1 \cdot a_2 \cdot . . . \cdot a_n + \sqrt{a_{n+1}-1})$ is positive integer

Call a number $n$ [i]good[/i] if it can be expressed as $2^x+y^2$ for where $x$ and $y$ are nonnegative integers. (a) Prove that there exist infinitely many sets of $4$ consecutive good numbers. (b) Find all sets of $5$ consecutive good numbers. [i]Proposed by Michael Ma[/i]

The $T_0$ set consists of all the numbers, representable as $(2k)!, k = 0, 1, 2, ... , n, ...$. The $T_m$ set is obtained from $T_{m-1}$ by adding all the finite sums of different numbers, that belong to $T_{m-1}$. Prove that there is a natural number, that doesn't belong to $T_{1987}$.

Find all positive integers \( m \) for which there exists an infinite subset \( A \) of the positive integers such that: for any pairwise distinct positive integers \( a_1, a_2, \cdots, a_m \in A \), the sum \( a_1 + a_2 + \cdots + a_m \) and the product \( a_1a_2 \cdots a_m \) are both square-free.