Prove that there exists two strictly increasing sequences $(a_{n})$ and $(b_{n})$ such that $a_{n}(a_{n}+1)$ divides $b^{2}_{n}+1$ for every natural n.

Say that a sequence $a_1$, $a_2$, $a_3$, $a_4$, $a_5$, $a_6$, $a_7$, $a_8$ is [i]cool[/i] if * the sequence contains each of the integers 1 through 8 exactly once, and * every pair of consecutive terms in the sequence are relatively prime. In other words, $a_1$ and $a_2$ are relatively prime, $a_2$ and $a_3$ are relatively prime, $\ldots$, and $a_7$ and $a_8$ are relatively prime. How many cool sequences are there?

Is there an infinite sequence of positive integers where no two terms are relatively prime, no term divides any other term, and there is no integer larger than $1$ that divides every term of the sequence?

Find all integer solutions $(x,y,z)$ of the equation $xy+yz+zx-xyz = 2$.

For a positive integer $M$, if there exist integers $a$, $b$, $c$ and $d$ so that: \[ M \leq a < b \leq c < d \leq M+49, \qquad ad=bc \] then we call $M$ a GOOD number, if not then $M$ is BAD. Please find the greatest GOOD number and the smallest BAD number.

Let $m$ be the smallest odd positive integer for which $1+ 2 +...+ m$ is a square of an integer and let $n$ be the smallest even positive integer for which $1 + 2 + ... + n$ is a square of an integer. What is the value of $m + n$?

Determine all three-digit numbers N such that $N^2$ has six digits and so that the sum of the number formed by the first three digits of $N^2$ and the number formed by the latter three digits of $N^2$ equals $N$.

Find all primes $p,q,r$ such that $qr-1$ is divisible by $p$, $pr-1$ is divisible by $q$, $pq-1$ is divisible by $r$.

Find all surjective functions $f: \mathbb{N}\rightarrow \mathbb{N}$ such that for all $m,n\in \mathbb{N}$ \[f(m)\mid f(n) \mbox{ if and only if }m\mid n.\]

Determine all pairs $(x, y)$ of positive integers satisfying $x + y + 1 | 2xy$ and $ x + y - 1 | x^2 + y^2 - 1$.

Let $n$ be a nonnegative integer, and let $p$ be a prime number that is congruent to $7$ modulo $8$. Prove that $$\sum_{k=1}^{p} \left\{ \frac{k^{2n}}{p} - \frac{1}{2} \right\} = \frac{p-1}{2}$$

Let be two real numbers $ \alpha ,\beta $ and two sequences $ \left(x_n \right)_{n\ge 1} ,\left(y_n \right)_{n\ge 1} $ whose smallest periods are $ p,q, $ respectively. Prove that the sequence $ \left( \alpha x_n+\beta y_n\right)_{n\ge 1} $ is periodic if $ \text{gcd}^2 (p,q) | \text{lcm} (p,q) , $ and in this case find its smallest period.

Find all positive integers which have exactly 16 positive divisors $1 = d_1 < d_2 < \ldots < d_{16} =n$ such that the divisor $d_k$, where $k = d_5$, equals $(d_2 + d_4) d_6$.

The largest one of numbers $ p_1^{\alpha_1}, p_2^{\alpha_2}, \cdots, p_t^{\alpha_t}$ is called a $ \textbf{Good Number}$ of positive integer $ n$, if $ \displaystyle n\equal{} p_1^{\alpha_1} \cdot p_2^{\alpha_2} \cdots p_t^{\alpha_t}$, where $ p_1$, $ p_2$, $ \cdots$, $ p_t$ are pairwisely different primes and $ \alpha_1, \alpha_2, \cdots, \alpha_t$ are positive integers. Let $ n_1, n_2, \cdots, n_{10000}$ be $ 10000$ distinct positive integers such that the $ \textbf{Good Numbers}$ of $ n_1, n_2, \cdots, n_{10000}$ are all equal. Prove that there exist integers $ a_1, a_2, \cdots, a_{10000}$ such that any two of the following $ 10000$ arithmetical progressions $ \{ a_i, a_i \plus{} n_i, a_i \plus{} 2n_i, a_i \plus{} 3n_i, \cdots \}$($ i\equal{}1,2, \cdots 10000$) have no common terms.

In a sequence $u_0,u_1,\ldots $ of positive integers, $u_0$ is arbitrary, and for any non-negative integer $n$, \[ u_{n+1}=\begin{cases}\frac{1}{2}u_n & \text{for even }u_n \\ a+u_n & \text{for odd }u_n \end{cases} \] where $a$ is a fixed odd positive integer. Prove that the sequence is periodic from a certain step.

Prove that there exist infinitely many positive integers $n$ such that the largest prime divisor of $n^4 + n^2 + 1$ is equal to the largest prime divisor of $(n+1)^4 + (n+1)^2 +1$.

Find all functions $f:\mathbb Z_{>0}\to \mathbb Z_{>0}$ such that $a+f(b)$ divides $a^2+bf(a)$ for all positive integers $a$ and $b$ with $a+b>2019$.

p1. From the measurement of the height of nine trees obtained data as following. a) There are three different measurement results (in meters) b) All data are positive numbers c) Mean$ =$ median $=$ mode $= 3$ d) The sum of the squares of all data is $87.$ Determine all possible heights of the nine trees. p2. If $x$ and $y$ are integers, find the number of pairs $(x,y)$ that satisfy $|x|+|y|\le 50$. p3. The plane figure $ABCD$ on the side is a trapezoid with $AB$ parallel to $CD$. Points $E$ and $F$ lie on $CD$ so that $AD$ is parallel to $BE$ and $AF$ is parallel to $BC$. Point $H$ is the intersection of $AF$ with $BE$ and point $G$ is the intersection of $AC$ with $BE$. If the length of $AB$ is $4$ cm and the length of $CD$ is $10$ cm, calculate the ratio of the area of ​​the triangle $AGH$ to the area of ​​the trapezoid $ABCD$. [img]https://cdn.artofproblemsolving.com/attachments/c/7/e751fa791bce62f091024932c73672a518a240.png[/img] p4. A prospective doctor is required to intern in a hospital for five days in July $2011$. The hospital leadership gave the following rules: a) Internships may not be conducted on two consecutive days. b) The fifth day of internship can only be done after four days counted since the fourth day of internship. Suppose the fourth day of internship is the date $20$, then the fifth day of internship can only be carried out at least the date $24$. Determine the many possible schedule options for the prospective doctor. p5. Consider the following sequences of natural numbers: $5$, $55$, $555$, $5555$, $55555$, $...$ ,$\underbrace{\hbox{5555...555555...}}_{\hbox{n\,\,numbers}}$ . The above sequence has a rule: the $n$th term consists of $n$ numbers (digits) $5$. Show that any of the terms of the sequence is divisible by $2011$.

* Given several numbers each of which is less than $1951$ and the least common multiple of any two of which is greater than $1951$. Prove that the sum of their reciprocals is less than $2$.

Compute the last two digits of $$9^{2020} + 9^{2020^2}+ ... + 9^{2020^{2020}}$$

Prove that for all positive integers $m$ and $n$ the following inequality hold: $$\pi(m)-\pi(n)\leq\frac{(m-1)\varphi(n)}{n}$$ When does equality hold? [i]Proposed by Shend Zhjeqi and Dorlir Ahmeti, Kosovo[/i]

Let $\mathbb{N}_{\geqslant 1}$ be the set of positive integers. Find all functions $f \colon \mathbb{N}_{\geqslant 1} \to \mathbb{N}_{\geqslant 1}$ such that, for all positive integers $m$ and $n$: \[\mathrm{GCD}\left(f(m),n\right) + \mathrm{LCM}\left(m,f(n)\right) = \mathrm{GCD}\left(m,f(n)\right) + \mathrm{LCM}\left(f(m),n\right).\] Note: if $a$ and $b$ are positive integers, $\mathrm{GCD}(a,b)$ is the largest positive integer that divides both $a$ and $b$, and $\mathrm{LCM}(a,b)$ is the smallest positive integer that is a multiple of both $a$ and $b$.

For fixed coprime positive integers $a,b$, define $n$ to be [i]bad[/i] if it is not of the form $$ax+by,\enspace x,y\in\mathbb N^*$$ Prove that there are finitely many bad positive integers. Also, find the sum of squares of them.

For each integer $n\geq 3$, find the least natural number $f(n)$ having the property $\star$ For every $A \subset \{1, 2, \ldots, n\}$ with $f(n)$ elements, there exist elements $x, y, z \in A$ that are pairwise coprime.

Suppose $ A $ is a subset of $ n $-elements taken from $ 1,2,3,4,...,2009 $ such that the difference of any two numbers in $ A $ is not a prime number. Find the largest value of $ n $ and the set $ A $ with this number of elements.