A list with $2007$ positive integers is written on a board, such that the arithmetic mean of all the numbers is $12$. Then, seven consecutive numbers are erased from the board. The arithmetic mean of the remaining numbers is $11.915$. The seven erased numbers have this property: the sixth number is half of the seventh, the fifth number is half of the sixth, and so on. Determine the $7$ erased numbers.

Let $ABCD$ be a square, such that the length of its sides are integers. This square is divided in $89$ smaller squares, $88$ squares that have sides with length $1$, and $1$ square that has sides with length $n$, where $n$ is an integer larger than $1$. Find all possible lengths for the sides of $ABCD$.

Let $ABCD$ be a square, $E$ and $F$ midpoints of $AB$ and $AD$ respectively, and $P$ the intersection of $CF$ and $DE$. a) Show that $DE \perp CF$. b) Determine the ratio $CF : PC : EP$

Each number from the set $\{1, 2, 3, 4, 5, 6, 7\}$ must be written in each circle of the diagram, so that the sum of any three aligned numbers is the same (e.g., $A+D+E = D+C+B$). What number cannot be placed on the circle $E$?

Let $A, B, C,$ be points in the plane, such that we can draw $3$ equal circumferences in which the first one passes through $A$ and $B$, the second one passes through $B$ and $C$, the last one passes through $C$ and $A$, and all $3$ circumferences share a common point $P$. Show that the radius of each of these circumferences is equal to the circumradius of triangle $ABC$, and that $P$ is the orthocenter of triangle $ABC$.