Let $M$ be a positive real number. Determine the least positive real number $k$ with the following property: for each integer $n>M$, the interval $(n,kn]$ contains a power of $2$. Authored by Nikola Velov

Let $ABCD$ be a quadrilateral with $AB>AD$ such that the inscribed circle $k_1$ of $\triangle ABC$ with center $O_1$ and the inscribed circle $k_2$ of $\triangle ADC$ with center $O_2$ have a common point on $AC$. If $k_1$ is tangent to $AB$ at $M$ and $k_2$ is tangent to $AD$ at $L$, prove that the lines $BD$, $LM$ and $O_1O_2$ pass through a common point.

Determine all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ which satisfy the equation $$f(f(x+y))=f(x+y)+f(x)f(y)-xy,$$for any two real numbers $x$ and $y$.

In two wooden boxes, there are $1994$ and $2024$ marbles, respectively. Spiro and Cvetko play the following game: alternately, each player takes a turn and removes some marbles from one of the boxes, so that the number of removed marbles in that turn is a divisor of the current number of marbles in the other box. The winner of the game is the one after whose turn both boxes are empty. Spiro takes the first turn. Which of the players has a winning strategy?

Let $f:\mathbb{N} \rightarrow \mathbb{N} \setminus \left \{ 1 \right \}$ be a function which satisfies both the inequality $f(a+f(a)) \leq 2a+3$ and the equation $$f(f(a)+b) = f(a+f(b)),$$for any two $a,b \in \mathbb{N}$. Let $g:\mathbb{N} \rightarrow \mathbb{N}$ be defined with: $g(a)$ is the largest prime divisor of $f(a)$. Prove that there exist integers $a>b>2024$ such that $b|a$ and $g(a) = g(b)$.