We say that a positive integer is a good number if the digit $2$ appears more often than the digit $3$ and that it is a bad number if the digit $3$ appears more often than the digit $2$. For example, $2023$ is a good number and $123$ is neither good nor bad. Calculate the difference between the quantity of good numbers and the quantity of bad numbers for integers less than or equal to $2023$.

Given the number $720$, Juan must choose $4$ numbers that are divisors of $720$. He wins if none of the four chosen numbers is a divisor of the product of the other three. Decide whether Juan can win.

In the parallelogram $ABCD$, the length of side $AB$ is half the length of side $BC$. The bisector of angle $\angle ABC$ intersects side $AD$ at point $K$ and diagonal $AC$ at point $L$. The bisector of angle $\angle ADC$ intersects the extension of side $AB$ at point $M$, with $B$ between $A$ and $M$. The line $ML$ intersects side $AD$ at point $F$. Calculate the ratio $\frac{AF}{AD}$.

Initially, Igna distributes $1000$ balls into $30$ boxes. Then, Igna and Mica alternate turns, starting with Igna. Each player, on their turn, chooses a box and removes one ball. When a player removes the last ball from a box, they earn a coin. Find the maximum integer $k$ such that, regardless of how Mica plays, Igna can earn at least $k$ coins.

A rectangular parallelepiped painted blue is cut into $1 \times 1\times 1$ cubes. Find the possible dimensions if the number of cubes without blue faces is equal to one-third of the total number of cubes. Note: A rectangular parallelepiped is a solid with $6$ faces, all of which are rectangles (or squares).

There is a row of $n$ chairs, numbered in order from left to right from $1$ to $n$. Additionally, the $n$ numbers from $1$ to $n$ are distributed on the backs of the chairs, one number per chair, such that the number on the back of a chair never matches the number of the chair itself. There is a child sitting on each chair. Every time the teacher claps, each child checks the number on the back of the chair they are sitting on and moves to the chair corresponding to that number. Prove that for any $m$ that is not a power of a prime, with $1 < m \leqslant n$, it is possible to distribute the numbers on the backrests such that, after the teacher claps $m$ times, for the first time, all the children are sitting in the chairs where they initially started. (During the process, it may happen that some children return to their original chairs, but they do not all do so simultaneously until the $m^{\text{th}}$ clap.)