A list of $2018$ numbers is created using the following procedure: the first number is $47$, the second number is $74$, and from there, each number is equal to the number formed by the last two digits of the sum of the two previous numbers:$$47, 74, 21, 95, 16, 11, \dots$$Bruno squares each of the $2018$ numbers and sums them. Determine the remainder when this sum is divided by $8$.

There are $n^2$ empty boxes, each with a square base. The height and width of each box are integers between $1$ and $n$ inclusive, and no two boxes are identical. One box fits inside another if its height and width are both smaller, and additionally, one of its dimensions is at least $2$ units smaller. In this way, we can form sequences of boxes (the first inside the second, the second inside the third, and so on). We place each of these sequences on a different shelf. How many shelves are needed to store all the boxes, with certainty?

A geometry program on the computer allows the following operations to be performed: Mark points on segments, on lines or outside them. Draw the line that joins two points. Find the point of intersection of two lines. Given a point $P$ and a line $\ell$, trace the symmetric of $P$ with respect to $\ell$. Given an triangle $ABC$, using exclusively the allowed operations, construct the intersection point of the perpendicular bisectors of the triangle.

There are $456$ people around a circle, denoted as $X_1, X_2, \dots, X_{456}$, and each one of them thought of a number. Every time Laura says an integer $k$ with $2 \leqslant k \leqslant 100$, the announcer announces all the numbers $p_1, p_2, \dots, p_{456}$, which are the averages of the numbers thought by the people in all the groups of $k$ consecutive people: $p_1$ is the average of the numbers thought by the people from $X_1$ to $X_k$, $p_2$ is the average of the numbers thought by the people from $X_2$ to $X_{k+1}$, and so on until $p_{456}$, the average of the numbers thought by the people from $X_{456}$ to $X_{k-1}$. Determine how many numbers $k$ Laura must say at a minimum so that, with certainty, the announcer can know the number thought by the person $X_{456}$.

A positive integer is called pretty if it is equal to the sum of the fourth powers of five distinct divisors. Prove that every pretty number is divisible by $5$. Determine if there are infinitely many beautiful numbers.

Ana writes a three-digit code, and Beto has to guess it. To do so, he can ask about a sequence of three digits, and Ana will respond "warm" if the sequence Beto proposes has at least one correct digit in the correct position, and she will respond "cold" if none of the digits are correct. For example, if the correct code is $014$, then if Beto asks $099$ or $014$, he receives the answer "warm", and if he asks $140$ or $322$, he receives the answer "cold". Determine the minimum number of questions Beto needs to ask in order to know the correct code with certainty.