Let $(F_{n})_{n\geq1}$ the Fibonacci sequence. Find all $n \in \mathbb{N}$ such that for every $k=0,1,...,F_{n}$ \[ {F_{n}\choose k} \equiv (-1)^{k} \ (mod \ F_{n}+1) \]

Let $\triangle ABC$ a scalene triangle with incenter $I$, circumcenter $O$ and circumcircle $\Gamma$. The incircle of $\triangle ABC$ is tangent to $BC, CA$ and $AB$ at points $D, E$ and $F$, respectively. The line $AI$ meet $EF$ and $\Gamma$ at $N$ and $M\neq A$, respectively. $MD$ meet $\Gamma$ at $L\neq M$ and $IL$ meet $EF$ at $K$. The circumference of diameter $MN$ meet $\Gamma$ at $P\neq M$. Prove that $AK, PN$ and $OI$ are concurrent.

In a mathematical challenge, positive real numbers $a_{1}\geq a_{2} \geq ... \geq a_{n}$ and an initial sequence of positive real numbers $(b_{1}, b_{2},...,b_{n+1})$ are given to Secco. Let $C$ a non-negative real number. In a sequence $(x_{1},x_{2},...,x_{n+1})$, consider the following operation: Subtract $1$ of some $x_{j}$, $j \in \{1,2,...,n+1\}$, add $C$ to $x_{n+1}$ and replace $(x_{1},x_{2},...,x_{j-1})$ for $(x_{1}+a_{\sigma (1)}, x_{2}+a_{\sigma (2)}, ..., x_{j-1}+a_{\sigma (j-1)})$, where $\sigma$ is a permutation of $(1,2,...,j-1)$. Secco's goal is to make all terms of sequence $(b_{k})$ negative after a finite number of operations. Find all values of $C$, depending of $a_{1}, a_{2},..., a_{n}, b_{1}, b_{2}, ..., b_{n+1}$, for which Secco can attain his goal.

Let $\triangle ABC$ an acute triangle of incenter $I$ and incircle $\omega$. $\omega$ is tangent to $BC, CA$ and $AB$ at points $T_{A}, T_{B}$ and $T_{C}$, respectively. Let $l_{A}$ the line through $A$ and parallel to $BC$ and define $l_{B}$ and $l_{C}$ analogously. Let $L_{A}$ the second intersection point of $AI$ with the circumcircle of $\triangle ABC$ and define $L_{B}$ and $L_{C}$ analogously. Let $P_{A}=T_{B}T_{C}\cap l_{A}$ and define $P_{B}$ and $P_{C}$ analogously. Let $S_{A}=P_{B}T_{B}\cap P_{C}T_{C}$ and define $S_{B}$ and $S_{C}$ analogously. Prove that $S_{A}L_{A}, S_{B}L_{B}, S_{C}L_{C}$ are concurrent.

Let $p$ a positive prime number and $\mathbb{F}_{p}$ the set of integers $mod \ p$. For $x\in \mathbb{F}_{p}$, define $|x|$ as the cyclic distance of $x$ to $0$, that is, if we represent $x$ as an integer between $0$ and $p-1$, $|x|=x$ if $x<\frac{p}{2}$, and $|x|=p-x$ if $x>\frac{p}{2}$ . Let $f: \mathbb{F}_{p} \rightarrow \mathbb{F}_{p}$ a function such that for every $x,y \in \mathbb{F}_{p}$ \[ |f(x+y)-f(x)-f(y)|<100 \]Prove that exist $m \in \mathbb{F}_{p}$ such that for every $x \in \mathbb{F}_{p}$ \[ |f(x)-mx|<1000 \]