Let $k\geqslant 2$ be an integer, and $a,b$ be real numbers. prove that $a-b$ is an integer divisible by $k$ if and only if for every positive integer $n$ $$\lfloor an \rfloor \equiv \lfloor bn \rfloor \ (mod \ k)$$ Proposed by Navid Safaei

Find all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ such that: $\bullet$ $f(x)<2$ for all $x\in (0,1)$; $\bullet$ for all real numbers $x,y$ we have: $$max\{f(x+y),f(x-y)\}=f(x)+f(y)$$ Proposed by Navid Safaei

Let $ABC$ be a scalene triangle with $I$ be its incenter. The incircle touches $BC$, $CA$, $AB$ at $D$, $E$, $F$, respectively. $Y$, $Z$ are the midpoints of $DF$, $DE$ respectively, and $S$, $V$ are the intersections of lines $YZ$ and $BC$, $AD$, respectively. $T$ is the second intersection of $\odot(ABC)$ and $AS$. $K$ is the foot from $I$ to $AT$. Prove that $KV$ is parallel to $DT$. Proposed by ltf0501

Given an acute triangle $ABC$, let $P$ be an arbitrary point on segment $BC$. A line passing through $P$ and perpendicular to $AC$ intersects $AB$ at $P_b$. A line passing through $P$ and perpendicular to $AB$ intersects $AC$ at $P_c$. Prove that the circumcircle of triangle $AP_bP_c$ passes through a fixed point other than $A$ when $P$ varies on segment $BC$. Proposed by ltf0501

Let $S$ be the set of lattice points whose both coordinates are positive integers no larger than $2022$. i.e., $S=\{(x, y) \mid x, y\in \mathbb{N}, \, 1\leq x, y\leq 2022\}$. We put a card with one gold side and one black side on each point in $S$. We call a rectangle "good" if: (i) All of its sides are parallel to the axes and have positive integer coordinates no larger than $2022$. (ii) The cards on its top-left and bottom-right corners are showing gold, and the cards on its top-right and bottom-left corners are showing black. Each "move" consists of choosing a good rectangle and flipping all cards simultaneously on its four corners. Find the maximum possible number of moves one can perform, or show that one can perform infinitely many moves. Proposed by CSJL

Find all completely multipiclative functions $f:\mathbb{Z}\rightarrow \mathbb{Z}_{\geqslant 0}$ such that for any $a,b\in \mathbb{Z}$ and $b\neq 0$, there exist integers $q,r$ such that $$a=bq+r$$and $$f(r)<f(b)$$ Proposed by Navid Safaei