In the cells of a $1 \times 100$ board, Julián writes all the integers from $1$ to $100$ (inclusive) in any order of his choice, without repeating numbers. For every three consecutive cells on the board, the cell containing the middle value of the three numbers in those cells is marked. For example, if the three numbers are $7$, $99$ and $22$, then the cell with $22$ is marked. Let $S$ be the sum of all the numbers in the marked cells. Determine the minimum value that $S$ can take. Note: Each marked number contributes to the sum $S$ exactly once, but it can be marked multiple times.

Point $D$ on the side $BC$ of the acute triangle $ABC$ is chosen so that $AD = AC$. Let $P$ and $Q$ respectively be the feet of the perpendiculars from $C$ and $D$ on the side $AB$. $AP^2 + 3BP^2 = AQ^2 + 3BQ^2$ is known. Calculate the measure of angle $\angle ABC$.

Nico wants to write the $100$ integers from $1$ to $100$ around a circle in some order and without repetition, such that they have the following property: when moving around the circle clockwise, the sum of the $100$ distances between each number and its next number is equal to $198$. Determine in how many ways the $100$ numbers can be ordered so that Nico achieves his goal. Note: The distance between two numbers $a$ and $b$ is equal to the absolute value of their difference: $|a - b|$.

There is a board with $n$ rows and $12$ columns. Each cell of the board contains a $1$ or a $0$. The board has the following properties: All rows are distinct. Each row contains exactly $4$ cells with $1$. For every $3$ rows, there is a column that intersects them in $3$ cells with $0$. Find the largest $n$ for which a board with these properties exists.

For each pair $a, \,b$ of coprime natural numbers, let $d_{a,\,b}$ be the greatest common divisor of $51a + b$ and $a + 51b$. Find the maximum possible value of $d_{a,\,b}$.

There are $999$ black points marked on a circle, dividing it into $999$ arcs of length $1$. We need to place $d$ arcs of lengths $1, 2, \dots, d$ such that each arc starts and ends at two black points, and none of the $d$ arcs is contained within another. Find the maximum value of $d$ for which this construction is possible. Note: Two arcs can have one or more black points in common.