Let $\mathbb{Z}_{>0}$ denote the set of positive integers. For any positive integer $k$, a function $f: \mathbb{Z}_{>0} \to \mathbb{Z}_{>0}$ is called [i]$k$-good[/i] if $\gcd(f(m) + n, f(n) + m) \le k$ for all $m \neq n$. Find all $k$ such that there exists a $k$-good function. [i]Proposed by James Rickards, Canada[/i]

Let \( n \) be a positive integer. A graph on \( 2n - 1 \) vertices is given such that the size of the largest clique in the graph is \( n \). Prove that there exists a vertex that is present in every clique of size \( n\)

The sequence of real numbers $a_1,a_2,...,a_{2015}$ is such that the 2015 equations: $a_1^3=a_1^2;a_1^3+a_2^3=(a_1+a_2 )^2;...;a_1^3+a_2^3+...+a_{2015}^3=(a_1+a_2+...+a_{2015} )^2$ are true. Prove that $a_1,a_2,…,a_{2015}$ are integers.

Find all functions $f: \mathbb{Z}\rightarrow\mathbb{Z}$ such that for all $x,y \in \mathbb{Z}$: \[f(x-y+f(y))=f(x)+f(y).\]

Let $k$ be a positive real. $A$ and $B$ play the following game: at the start, there are $80$ zeroes arrange around a circle. Each turn, $A$ increases some of these $80$ numbers, such that the total sum added is $1$. Next, $B$ selects ten consecutive numbers with the largest sum, and reduces them all to $0$. $A$ then wins the game if he/she can ensure that at least one of the number is $\geq k$ at some finite point of time. Determine all $k$ such that $A$ can always win the game.

Let $a_1,a_2,\ldots a_n,k$, and $M$ be positive integers such that $$\frac{1}{a_1}+\frac{1}{a_2}+\cdots+\frac{1}{a_n}=k\quad\text{and}\quad a_1a_2\cdots a_n=M.$$ If $M>1$, prove that the polynomial $$P(x)=M(x+1)^k-(x+a_1)(x+a_2)\cdots (x+a_n)$$ has no positive roots.

Let $x_1,x_2,x_3, \dots$ be a sequence of nonzero real numbers satisfying $$x_n=\frac{x_{n-2}x_{n-1}}{2x_{n-2}-x_{n-1}} \text{ for } n=3,4,5, \dots.$$ Establish necessary and sufficient conditions on $x_1$ and $x_2$ for $x_n$ to be an integer for infinitely many values of $n.$

In a $n\times n$ table ($n>1$) $k$ unit squares are marked.One wants to rearrange rows and columns so that all the marked unit squares are above the main diagonal or on it.For what maximum $k$ is it always possible?

Let $ n$ be an even positive integer. Prove that there exists a positive inter $ k$ such that \[ k \equal{} f(x) \cdot (x\plus{}1)^n \plus{} g(x) \cdot (x^n \plus{} 1)\] for some polynomials $ f(x), g(x)$ having integer coefficients. If $ k_0$ denotes the least such $ k,$ determine $ k_0$ as a function of $ n,$ i.e. show that $ k_0 \equal{} 2^q$ where $ q$ is the odd integer determined by $ n \equal{} q \cdot 2^r, r \in \mathbb{N}.$ Note: This is variant A6' of the three variants given for this problem.

[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 $n$ be a positive integer. Given is a subset $A$ of $\{0,1,...,5^n\}$ with $4n+2$ elements. Prove that there exist three elements $a<b<c$ from $A$ such that $c+2a>3b$. [i]Proposed by Dominik Burek and Tomasz Ciesla, Poland[/i]

Let $G$ be a directed graph with infinitely many vertices. It is known that for each vertex the outdegree is greater than the indegree. Let $O$ be a fixed vertex of $G$. For an arbitrary positive number $n$, let $V_{n}$ be the number of vertices which can be reached from $O$ passing through at most $n$ edges ( $O$ counts). Find the smallest possible value of $V_{n}$.

There is a unique sequence of integers $a_1, a_2, \cdots a_{2023}$ such that $$ \tan2023x = \frac{a_1 \tan x + a_3 \tan^3 x + a_5 \tan^5 x + \cdots + a_{2023} \tan^{2023} x}{1 + a_2 \tan^2 x + a_4 \tan^4 x \cdots + a_{2022} \tan^{2022} x} $$ whenever $\tan 2023x$ is defined. What is $a_{2023}?$ $\textbf{(A) } -2023 \qquad\textbf{(B) } -2022 \qquad\textbf{(C) } -1 \qquad\textbf{(D) } 1 \qquad\textbf{(E) } 2023$

Determine all functions $f\colon\mathbb{Z}_{>0}\to\mathbb{Z}_{>0}$ such that, for all positive integers $a$ and $b$, \[ f^{bf(a)}(a+1)=(a+1)f(b). \]

There are $n \ge 2$ numbers on the blackboard: $1, 2,..., n$. It is permitted to erase two of those numbers $x,y$ and write $2x - y$ instead. Find all values of $n$ such that it is possible to leave number $0$ on the blackboard after $n - 1$ such procedures.