We call a set “sum free” if no two elements of the set add up to a third element of the set. What is the maximum size of a sum free subset of $\{ 1, 2, \ldots , 2n - 1 \}$.

Proof that $$ \sum_{m=1}^n5^{\omega (m)} \le \sum_{k=1}^n\lfloor \frac{n}{k} \rfloor \tau (k)^2 \le \sum_{m=1}^n5^{\Omega (m)} .$$

Determine whether there exists a function $f: \mathbb{Z}_{> 0} \rightarrow \mathbb{Z}_{> 0}$ such that for all positive integers $m$ and $n$, \[f(m+nf(m))=f(n)^m+2024! \cdot m.\] [i]Jaedon Whyte[/i]

Let $n \geq 2$ be a positive integer. A total of $2n$ balls are coloured with $n$ colours so that there are two balls of each colour. These balls are put inside $n$ cylindrical boxes with two balls in each box, one on top of the other. Phoe Wa Lone has an empty cylindrical box and his goal is to sort the balls so that balls of the same colour are grouped together in each box. In a [i]move[/i], Phoe Wa Lone can do one of the following: [list] [*]Select a box containing exactly two balls and reverse the order of the top and the bottom balls. [*]Take a ball $b$ at the top of a non-empty box and either put it in an empty box, or put it in the box only containing the ball of the same colour as $b$. [/list] Find the smallest positive integer $N$ such that for any initial placement of the balls, Phoe Wa Lone can always achieve his goal using at most $N$ moves in total.

Let $a_1, a_2, a_3, \ldots$ be a sequence of positive real numbers, and $s$ be a positive integer, such that \[a_n = \max \{ a_k + a_{n-k} \mid 1 \leq k \leq n-1 \} \ \textrm{ for all } \ n > s.\] Prove there exist positive integers $\ell \leq s$ and $N$, such that \[a_n = a_{\ell} + a_{n - \ell} \ \textrm{ for all } \ n \geq N.\] [i]Proposed by Morteza Saghafiyan, Iran[/i]

Let $A$ be a finite set of non-negative integers. Determine all functions $f:\mathbb{Z}_{\ge 0} \to A$ such that \[f(|x-y|)=|f(x)-f(y)|\] for each $x,y\in\mathbb Z_{\ge 0}$. [i]Andrei Bâra[/i]

On competition which has $16$ teams, it is played $55$ games. Prove that among them exists $3$ teams such that they have not played any matches between themselves.

Let a and b be non-negative integers such that $ab \ge c^{2}$ where $c$ is an integer. Prove that there is a positive integer n and integers $x_{1}$, $x_{2}$, $\cdots$, $x_{n}$, $y_{1}$, $y_{2}$, $\cdots$, $y_{n}$ such that \[{x_{1}}^{2}+\cdots+{x_{n}}^{2}=a,\;{y_{1}}^{2}+\cdots+{y_{n}}^{2}=b,\; x_{1}y_{1}+\cdots+x_{n}y_{n}=c\]

Let $d$ be a real number. For each integer $m \geq 0,$ define a sequence $\left\{a_{m}(j)\right\}, j=0,1,2, \ldots$ by the condition \begin{align*} a_{m}(0)&=d / 2^{m},\\ a_{m}(j+1)&=\left(a_{m}(j)\right)^{2}+2 a_{m}(j), \quad j \geq 0. \end{align*} Evaluate $\lim _{n \rightarrow \infty} a_{n}(n).$

Find all strictly increasing functions $f: \mathbb{N}\to \mathbb{N}$ such that \[f(f(n))=3n.\]

A rectangle $\mathcal R$ is divided into a set $\mathcal S$ of finitely many smaller rectangles with sides parallel to the sides of $\mathcal R$ such that no three rectangles in $\mathcal S$ share a common corner. An ant is initially located at the bottom-left corner of $\mathcal R$. In one operation, we can choose a rectangle $r$ in $\mathcal S$ such that the ant is currently located at one of the corners of $r$, say $c$, and move the ant to one of the two corners of $r$ adjacent to $c$. Suppose that after a finite number of operations, the ant ends up at the top-right corner of $\mathcal R$. Prove that some rectangle $r$ in $\mathcal S$ was chosen in at least two operations.

Let $p$ be a prime. Prove that any complete graph with $1000p$ vertices, whose edges are labelled with integers, has a cycle whose sum of labels is divisible by $p$.

Let $(a_n)_{n\geq 1}$ be a sequence of positive real numbers with the property that $$(a_{n+1})^2 + a_na_{n+2} \leq a_n + a_{n+2}$$ for all positive integers $n$. Show that $a_{2022}\leq 1$.

Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $f(yf(x))+f(x-1)=f(x)f(y)$ and $|f(x)|<2022$ for all $0<x<1$.

A sequence $(a_n)_{n\ge0}$ satisfies $a_{m+n}+a_{m-n}=\frac12\left(a_{2m}+a_{2n}\right)$ for all integers $m,n$ with $m\ge n\ge0$. Given that $a_1=1$, find $a_{2003}$.