Victor writes down all $7-$digit numbers using the digits $1, 2, 3, 4, 5, 6,$ and $7$ exactly once. Prove that there are no two numbers among them where one is a multiple of the other.

Let $ABC$ be an acute-angled triangle, $D$ be the foot of the altitude from $A$, the circle with diameter $AD$ intersect $AB$ at $F$ and $AC$ at $E$. Let $P$ be the orthocenter of triangle $AEF$ and $O$ be the circumcenter of $ABC$. Prove that $A, P,$ and $O$ are collinear.

Let $n$ be a positive integer. A grid of $n\times n$ has some black-colored cells. Drini can color a cell if at least three cells that share a side with it are also colored black. Drini discovers that by repeating this process, all the cells in the grid can be colored. Prove that if there are initially $k$ colored cells, then $$k\geq \frac{n^2+2n}{3}.$$

Given the Fibonacci sequence with $f_0=f_1=1$and for $n\geq 1, f_{n+1}=f_n+f_{n-1}$, find all real solutions to the equation: $$x^{2024}=f_{2023}x+f_{2022}.$$