Find the smallest positive integer $n$ or show no such $n$ exists, with the following property: there are infinitely many distinct $n$-tuples of positive rational numbers $(a_1, a_2, \ldots, a_n)$ such that both $$a_1+a_2+\dots +a_n \quad \text{and} \quad \frac{1}{a_1} + \frac{1}{a_2} + \dots + \frac{1}{a_n}$$ are integers.

We define the following sequence: $a_0=a_1=1$, $a_{n+1}=14a_n-a_{n-1}$. Prove that $2a_n-1$ is a perfect square.

How many integer solutions exist that satisfy this equation? $$x+4y-343\sqrt{x}-686\sqrt{y}+4\sqrt{xy}+2022=0$$.

Determine all triples $(x, y, z)$ of positive integers $x > y > z > 0$, such that $x^2 = y \cdot 2^z + 1$

Find all triples of natural numbers $ a $, $ b $, $ c $ such that $$ \gcd (a ^ 2, b ^ 2) + \gcd (a, bc) +\gcd (b, ac) +\gcd (c, ab) = 239 ^ 2 = ab + c . $$

[b]p1.[/b] There are eight different dominoes in the box (fig.), but the boundaries between them are not visible. Draw the boundaries. [img]https://cdn.artofproblemsolving.com/attachments/6/f/6352b18c25478d68a23820e32a7f237c9f2ba9.png[/img] [b]p2.[/b] The teacher drew a quadrilateral $ABCD$ on the board. Vanya and Vitya marked points $X$ and $Y$ inside it, from which all sides of the quadrilateral are visible at equal angles. What is the distance between points $X$ and $Y$? (From point $X$, side $AB$ is visible at angle $AXB$.) [b]pЗ.[/b] Several identical black squares, perhaps partially overlapping, were placed on a white plane. The result was a black polygonal figure, possibly with holes or from several pieces. Could it be that this figure does not have a single right angle? [b]p4.[/b] The bus ticket number consists of six digits (the first digits may be zeros). A ticket is called [i]lucky [/i] if the sum of the first three digits is equal to the sum of the last three. Prove that the sum of the numbers of all lucky tickets is divisible by $13$. [b]p5.[/b] The Meandrovka River, which has many bends, crosses a straight highway under thirteen bridges. Prove that there are two neighboring bridges along both the highway and the river. (Bridges are called river neighbors if there are no other bridges between them on the river section; bridges are called highway neighbors if there are no other bridges between them on the highway section.) PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics]here.[/url]

Let $\mathbb{N}$ be the set of all positive integers. Find all functions $f : \mathbb{N} \rightarrow \mathbb{N}$ such that $f(x) + y$ and $f(y) + x$ have the same number of $1$'s in their binary representations, for any $x,y \in \mathbb{N}$.

[b]p1.[/b] A function $f$ defined on the set of positive numbers satisfies the equality $$f(xy) = f(x) + f(y), x, y > 0.$$ Find $f(2007)$ if $f\left( \frac{1}{2007} \right) = 1$. [b]p2.[/b] The plane is painted in two colors. Show that there is an isosceles right triangle with all vertices of the same color. [b]p3.[/b] Show that the number of ways to cut a $2n \times 2n$ square into $1\times 2$ dominoes is divisible by $2$. [b]p4.[/b] Two mirrors form an angle. A beam of light falls on one mirror. Prove that the beam is reflected only finitely many times (even if the angle between mirrors is very small). [b]p5.[/b] A sequence is given by the recurrence relation $a_{n+1} = (s(a_n))^2 +1$, where $s(x)$ is the sum of the digits of the positive integer $x$. Prove that starting from some moment the sequence is periodic. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Set $p\ge 5$ be a prime number and $n$ be a natural number. Let $f$ be a function $ f: \mathbb{Z_{ \neq }}_0 \rightarrow \mathbb{ N }_0 $ satisfy the following conditions: i) For all sequences of integers satisfy $ a_i \not\in \{0, 1\} $, and $ p $ $\not |$ $ a_i-1 $, $ \forall $ $ 1 \le i \le p-2 $,\\ $$ \displaystyle \sum^{p-2}_{i=1}f(a_i)=f(a_1a_2 \cdots a_{p-2}) $$ ii) For all coprime integers $ a $ and $ b $, $ a \equiv b \pmod p \Rightarrow f(a)=f(b) $ iii) There exist $k \in \mathbb{Z}_{\neq 0} $ that satisfy $ f(k)=n $ Prove that the number of such functions is $ d(n) $, where $ d(n) $ denotes the number of divisors of $ n $.

Find the sum of all integers $m$ with $1 \le m \le 300$ such that for any integer $n$ with $n \ge 2$, if $2013m$ divides $n^n-1$ then $2013m$ also divides $n-1$. [i]Proposed by Evan Chen[/i]

Find the largest positive integer with the property that each digit apart from the first and the last one is smaller than the arithmetic mean of her neighbours.

For every positive integer $n$, show that there is a positive integer $k$ such that $2k^2 + 2001k + 3 \equiv 0$ (mod $2^n$).

Let $n\geq 2$ be a positive integer. Prove that there exists a positive integer $m$, such that $n\mid m, \ m<n^4$ and at most four distinct digits are used in the decimal representation of $m$.

Find all pairs of prime numbers $(p, q)$ for which the numbers $p+q$ and $p+4q$ are simultaneously perfect squares.

Let $ a$, $ b$ and $ m$ be positive integers. Prove that there exists a positive integer $ n$ such that $ (a^n \minus{} 1)b$ is divisible by $ m$ if and only if $ \gcd (ab, m) \equal{} \gcd (b, m)$.

Suppose that $ y \equal{} \frac34x$ and $ x^y \equal{} y^x$. The quantity $ x \plus{} y$ can be expressed as a rational number $ \frac{r}{s}$, where $ r$ and $ s$ are relatively prime positive integers. Find $ r \plus{} s$.

Find the integer solutions of the equation \[ \left[ \sqrt{2m} \right] = \left[ n(2+\sqrt 2) \right] \]

Let $n$ be a positive integer, and let $a_1,a_2,..,a_n$ be pairwise distinct positive integers. Show that $$\sum_{k=1}^{n}{\frac{1}{[a_1,a_2,…,a_k]}} <4,$$ where $[a_1,a_2,…,a_k]$ is the least common multiple of the integers $a_1,a_2,…,a_k$.

Find all prime numbers $p$ and $q$ such that $$(2p-q)^2=17p-10q$$

Arthur, Bob, and Carla each choose a three-digit number. They each multiply the digits of their own numbers. Arthur gets 64, Bob gets 35, and Carla gets 81. Then, they add corresponding digits of their numbers together. The total of the hundreds place is 24, that of the tens place is 12, and that of the ones place is 6. What is the difference between the largest and smallest of the three original numbers? [i]Proposed by Jacob Weiner[/i]

Find all pairs ($x,y$) of integers such that $x^2-2xy+126y^2=2009$.

Phoenix is counting positive integers starting from $1$. When he counts a perfect square greater than $1$, he restarts at $1$, skipping that square the next time. For example, the first $10$ numbers Phoenix counts are $1$, $2$, $3$, $4$, $1$, $2$, $3$, $5$, $6$, $7$, $...$ How many numbers will Phoenix have counted after counting 1$00$ for the first time?

Find all positive integers $a, b, c$ such that $ab+1$ and $c$ are coprimes and: $$a(ba+1)(ca^2+ba+1)=2021^{2021}$$

Let $S$ be a set of positive integers, such that $n \in S$ if and only if $$\sum_{d|n,d<n,d \in S} d \le n$$ Find all positive integers $n=2^k \cdot p$ where $k$ is a non-negative integer and $p$ is an odd prime, such that $$\sum_{d|n,d<n,d \in S} d = n$$

Prove that, for all integers $a, b$, there exists a positive integer $n$, such that the number $n^2+an+b$ has at least $2018$ different prime divisors.