Let $n \geqslant 3$ be an integer. A frog is to jump along the real axis, starting at the point 0 and making $n$ jumps: one of length 1 , one of length $2, \ldots$, one of length $n$. It may perform these $n$ jumps in any order. If at some point the frog is sitting on a number $a \leqslant 0$, its next jump must be to the right (towards the positive numbers). If at some point the frog is sitting on a number $a>0$, its next jump must be to the left (towards the negative numbers). Find the largest positive integer $k$ for which the frog can perform its jumps in such an order that it never lands on any of the numbers $1,2, \ldots, k$.

Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that

$$f(x+y)+y \leqslant f(f(f(x)))$$

holds for all $x, y \in \mathbb{R}$.

Let $\triangle A B C$ be a triangle with circumcircle $\Gamma$, and let $I$ be the center of the incircle of $\triangle A B C$. The lines $A I, B I$ and $C I$ intersect $\Gamma$ in $D \neq A, E \neq B$ and $F \neq C$. The tangent lines to $\Gamma$ in $F, D$ and $E$ intersect the lines $A I, B I$ and $C I$ in $R, S$ and $T$, respectively. Prove that

$$|A R| \cdot|B S| \cdot|C T|=|I D| \cdot|I E| \cdot|I F|$$

a) Find all positive integers $g$ with the following property: for each odd prime number $p$ there exists a positive integer $n$ such that $p$ divides the two integers

$$g^{n}-n \text { and } g^{n+1}-(n+1)$$

b) Find all positive integers $g$ with the following property: for each odd prime number $p$ there exists a positive integer $n$ such that $p$ divides the two integers

$$g^{n}-n^{2} \text { and } g^{n+1}-(n+1)^{2}$$