Let $n \ge 2$ be an integer and let $a_1, a_2, \cdots a_n$ be fixed positive integers (not necessarily all distinct) in such a way that $\gcd(a_1, a_2 \cdots a_n)=1$. In a board the numbers $a_1, a_2 \cdots a_n$ are all written along with a positive integer $x$. A move consists of choosing two numbers $a>b$ from the $n+1$ numbers in the board and replace them with $a-b,2b$. Find all possible values of $x$, with respect of the values of $a_1, a_2 \cdots a_n$, for which it is possible to achieve a finite sequence of moves (possibly none) such that eventually all numbers written in the board are equal.

Solve the equation $$x^2+y^2+z^2=686$$ where $x$, $y$ and $z$ are positive integers

Ann wrote down all the perfect squares from one to one million (all in a single line). However, at night, an evil elf erased one of the numbers. So the next day, Ann saw an empty space between the numbers $760384$ and $763876$. What is the sum of the digits of the erased number?

Decide whether there exists a set $M$ of positive integers satisfying the following conditions: (i) For any natural number $m>1$ there exist $a, b \in M$ such that $a+b = m.$ (ii) If $a, b, c, d \in M$, $a, b, c, d > 10$ and $a + b = c + d$, then $a = c$ or $a = d.$

Given integers $a$, $ b$ and $c$, $c\ne b$. It is known that the square trinomials $ax^2 + bx + c$ and $(c-b)x^2 + (c- a)x + (a + b)$ have a common root (not necessarily integer). Prove that $a+b+2c$ is divisible by $3$.

Find the number of polynomials $f(x)=ax^3+bx$ satisfying both following conditions: (i) $a,b\in\{1,2,\ldots,2013\}$; (ii) the difference between any two of $f(1),f(2),\ldots,f(2013)$ is not a multiple of $2013$.

Let $P(x)$ be a polynomial with integer coefficients. We denote $\deg(P)$ its degree which is $\geq 1.$ Let $n(P)$ be the number of all the integers $k$ for which we have $(P(k))^{2}=1.$ Prove that $n(P)- \deg(P) \leq 2.$

For positive integer $x>1$, define set $S_x$ as $$S_x=\{p^\alpha|p \textup{ is one of the prime divisor of }x,\alpha \in \mathbb{N},p^\alpha|x,\alpha \equiv v_p(x)(\textup{mod} 2)\},$$ where $v_p(n)$ is the power of prime divisor $p$ in positive integer $n.$ Let $f(x)$ be the sum of all the elements of $S_x$ when $x>1,$ and $f(1)=1.$ Let $m$ be a given positive integer, and the sequence $a_1,a_2,\cdots,a_n,\cdots$ satisfy that for any positive integer $n>m,$ $a_{n+1}=\max\{ f(a_n),f(a_{n-1}+1),\cdots,f(a_{n-m}+m)\}.$ Prove that (1)there exists constant $A,B(0<A<1),$ such that when positive integer $x$ has at least two different prime divisors, $f(x)<Ax+B$ holds; (2)there exists positive integer $Q$, such that for any positive integer $n,a_n<Q.$

[u]Round 5[/u] [b]p13.[/b] For a square pyramid whose base has side length $9$, a square is formed by connecting the centroids of the four triangular faces. What is the area of the square formed by the centroids? [b]p14.[/b] Farley picks a real number p uniformly at random in the range $\left( \frac13, \frac23 \right)$. She then creates a special coin that lands on heads with probability $p$ and tails with probability $1 - p$. She flips this coin, and it lands on heads. What is the probability that $p > \frac12$? [b]p15.[/b] Let $ABCD$ be a quadrilateral with $\angle A = \angle C = 90^o$. Extend $AB$ and $CD$ to meet at point $P$. Given that $P B = 3$, $BA = 21$, and $P C = 1$, find $BD^2$ [u]Round 6[/u] [b]p16.[/b] Three congruent, mutually tangent semicircles are inscribed in a larger semicircle, as shown in the diagram below. If the larger semicircle has a radius of $30$ units, what is the radius of one of the smaller semicircles? [img]https://cdn.artofproblemsolving.com/attachments/5/e/1b73791e95dc4ed6342f0151f3f63e1b31ae3c.png[/img] [b]p17.[/b] In isosceles trapezoid $ABCD$ with $BC \parallel AD$, the distances from $A$ and $B$ to line $CD$ are $3$ and $9$, respectively. If the distance between the two bases of trapezoid $ABCD$ is $5$, find the area of quadrilateral $ABCD$. [b]p18.[/b] How many ways are there to tile the “$E$” shape below with dominos? A domino covers two adjacent squares. [img]https://cdn.artofproblemsolving.com/attachments/b/b/82bdb8d8df8bc3d00b9aef9eb39e55358c4bc6.png[/img] [u]Round 7[/u] [b]p19.[/b] In isoceles triangle $ABC$, $AC = BC$ and $\angle ACB = 20^o$. Let $\Omega$ be the circumcircle of triangle $ABC$ with center $O$, and let $M$ be the midpoint of segment $BC$. Ray $\overrightarrow{OM}$ intersects $\Omega$ at $D$. Let $\omega$ be the circle with diameter $OD$. $AD$ intersects $\omega$ again at a point $X$ not equal to $D$. Given $OD = 2$, find the area of triangle $OXD$. [b]p20.[/b] Find the smallest odd prime factor of $2023^{2029} + 2026^{2029} - 1$. [b]p21.[/b] Achyuta, Alan, Andrew, Anish, and Ava are playing in the EMCC games. Each person starts with a paper with their name taped on their back. A person is eliminated from the game when anybody rips their paper off of their back. The game ends when one person remains. The remaining person then rips their paper off of their own back. At the end of the game, each person collects the papers that they ripped off. How many distinct ways can the papers be distributed at the end of the game? [u]Round 8[/u] [b]p22.[/b] Anthony has three random number generators, labelled $A$, $B$ and $C$. $\bullet$ Generator$ A$ returns a random number from the set $\{12, 24, 36, 48, 60\}$. $\bullet$ Generator $B$ returns a random number from the set $ \{15, 30, 45, 60\}$. $\bullet$ Generator $C$ returns a random number from the set $\{20, 40, 60\}$. He uses generator $A$, $B$, and then $C$ in succession, and then repeats this process indefinitely. Anthony keeps a running total of the sum of all previously generated numbers, writing down the new total every time he uses a generator. After he uses each machine $10 $ times, what is the average number of multiples of $60$ that Anthony will have written down? [b]p23.[/b] A laser is shot from one of the corners of a perfectly reflective room shaped like an equilateral triangle. The laser is reflected 2497 times without shining into a corner of the room, but after the 2497th reflection, it shines directly into the corner it started from. How many different angles could the laser have been initially pointed? [b]p24.[/b] We call a k-digit number blissful if the number of positive integers $n$ such that $n^n$ ends in that $k$-digit number happens to be nonzero and finite. What is the smallest value of $k$ such that there exists a blissful $k$-digit number? PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h3131523p28369592]here[/url].. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $x,y$ be the positive integers such that $3x^2 +x = 4y^2 + y$. Prove that $x - y$ is a perfect (square).

The function $ f : \mathbb{N} \to \mathbb{Z}$ is defined by $ f(0) \equal{} 2$, $ f(1) \equal{} 503$ and $ f(n \plus{} 2) \equal{} 503f(n \plus{} 1) \minus{} 1996f(n)$ for all $ n \in\mathbb{N}$. Let $ s_1$, $ s_2$, $ \ldots$, $ s_k$ be arbitrary integers not smaller than $ k$, and let $ p(s_i)$ be an arbitrary prime divisor of $ f\left(2^{s_i}\right)$, ($ i \equal{} 1, 2, \ldots, k$). Prove that, for any positive integer $ t$ ($ t\le k$), we have $ 2^t \Big | \sum_{i \equal{} 1}^kp(s_i)$ if and only if $ 2^t | k$.

Find all prime numbers $ p$ and $ q$ such that $ p^3 \minus{} q^5 \equal{} (p \plus{} q)^2$.

Determine the sum of all positive integers $n$ between $1$ and $100$ inclusive such that \[\gcd(n,2^n - 1) = 3.\]

Say a positive integer $n>1$ is $d$-coverable if for each non-empty subset $S\subseteq \{0, 1, \ldots, n-1\}$, there exists a polynomial $P$ with integer coefficients and degree at most $d$ such that $S$ is exactly the set of residues modulo $n$ that $P$ attains as it ranges over the integers. For each $n$, find the smallest $d$ such that $n$ is $d$-coverable, or prove no such $d$ exists. [i]Proposed by Carl Schildkraut[/i]

p is a prime number, k is a positive integer Find all (p, k): $k!=(p^3-1)(p^3-p)(p^3-p^2)$

Let $ a_1,a_2,a_3,\dots$ be infinite sequence of positive integers satisfying the following conditon: for each prime number $ p$, there are only finite number of positive integers $ i$ such that $ p|a_i$. Prove that that sequence contains a sub-sequence $ a_{i_1},a_{i_2},a_{i_3},\dots$, with $ 1 \le i_1<i_2<i_3<\dots$, such that for each $ m \ne n$, $ \gcd(a_{i_m},a_{i_n})\equal{}1$.

Let $S(n)$ denote the sum of digits of $n \in N$. Find all $n$ such that $S(n) = S(2n) = S(3n) =... = S(n^2)$

Is $$\prod_{k=0}^\infty \left(1-\frac{1}{2022^{k!}}\right)$$ rational?

Find the least positive integer $n$ satisfying the following statement: for eash pair of positive integers $a$ and $b$ such that $36$ divides $a+b$ and $n$ divides $ab$ it follows that $36$ divides both $a$ and $b$.

Determine all pairs $(x, y)$ of positive integers such that \[\sqrt[3]{7x^2-13xy+7y^2}=|x-y|+1.\] [i]Proposed by Titu Andreescu, USA[/i]

A positive integer $k\geqslant 3$ is called[i] fibby[/i] if there exists a positive integer $n$ and positive integers $d_1 < d_2 < \ldots < d_k$ with the following properties: \\ $\bullet$ $d_{j+2}=d_{j+1}+d_j$ for every $j$ satisfying $1\leqslant j \leqslant k-2$, \\ $\bullet$ $d_1, d_2, \ldots, d_k$ are divisors of $n$, \\ $\bullet$ any other divisor of $n$ is either less than $d_1$ or greater than $d_k$. Find all fibby numbers. \\ \\ [i]Proposed by Ivan Novak.[/i]

Let $k$ be a nonzero natural number and $m$ an odd natural number . Prove that there exist a natural number $n$ such that the number $m^n+n^m$ has at least $k$ distinct prime factors.

Let $x$ and $y$ be positive integers such that $xy$ divides $x^{2}+y^{2}+1$. Show that \[\frac{x^{2}+y^{2}+1}{xy}=3.\]

Prove that the rectangle $m\times n$ table can be filled with exact squares so, that the sums in the rows and the sums in the columns will be exact squares also.

Let $a,b,c,d$ be pairwise distinct positive integers such that $$\frac{a}{a+b}+\frac{b}{b+c}+\frac{c}{c+d}+\frac{d}{d+a}$$ is an integer. Prove that $a+b+c+d$ is [b]not[/b] a prime number.