We say that three positive integers $a$, $b$ and $c$ form a family if the following two conditions are satisfied: $a + b + c = 900$. There exists an integer $n$, with $n \geqslant 2$, such that $$\frac{a}{n-1}=\frac{b}{n}=\frac{c}{n+1}.$$ Determine the number of such families.

A $7 \times 7$ grid is given. Julián colors $29$ cells black. Pilar must then place an $L$-shaped piece, covering exactly three cells (oriented in any direction, as shown in the figure). Pilar wins if all three cells covered by the $L$-shaped piece are black. Can Julián color the grid in such a way that it is impossible for Pilar to win? [asy][asy] size(1.5cm); draw((0,1)--(1,1)--(1,2)--(0,2)--(0,1)--(0,0)--(1,0)--(2,0)--(2,1)--(1,1)--(1,0)); [/asy][/asy]

Let $\Gamma$ be a circle of center $S$ and radius $r$ and $A$ a point outside the circle. Let $BC$ be a diameter of $\Gamma$ such that $B$ does not belong to the line $AS$, and we consider the point $O$ where the perpendicular bisectors of triangle $ABC$ intersect, that is, the circumcenter of $ABC$. Determine all possible locations of point $O$ when $B$ varies in circle $\Gamma$.

We define similar numbers as positive integers that have exactly the same digits (but possibly in another order). For example, $1241$, $2114$ and $4211$ are similar numbers, but $1424$ is not similar to the other three. Determine whether there exist three similar numbers, each with $300$ digits (all digits being non-zero), such that the sum of two of them equals the third. If the answer is yes, provide an example; if not, justify why it is impossible.

In a club, some pairs of members are friends. Given an integer $k \geqslant 3$, we say a club is $k$-friendly if, in any group of $k$ members, they can be seated at a round table such that each pair of neighbors are friends. Prove that if a club is $6$-friendly, then it is also $7$-friendly. Is it true that if a club is $9$-friendly, then it is also $10$-friendly?

Let $n$ be a natural number. We define $f(n)$ as the number of ways to express $n$ as a sum of powers of $2$, where the order of the terms is taken into account. For example, $f(4) = 6$, because $4$ can be written as: \begin{align*} 4;\\ 2 + 2;\\ 2 + 1 + 1;\\ 1 + 2 + 1;\\ 1 + 1 + 2;\\ 1 + 1 + 1 + 1. \end{align*}Find the smallest $n$ greater than $2019$ for which $f(n)$ is odd.