In the following sequence of numbers, each term, starting with the third, is obtained by adding three times the previous term plus twice the previous term to the previous one: $$a_1, a_2, 78, a_4, a_5, 3438, a_7, a_8,…$$As seen in the sequence, the third term is $78$ and the sixth term is $3438$. What is the value of the term $a_7$?

Aidée draws ten squares of different sizes. The diagonal of the first square measures $1$ cm, the diagonal of the second measures $2$ cm, the diagonal of the third measures $3$ cm, and so on until the diagonal of the tenth square measures $10$ cm. How much are the areas of the ten squares?

In the figure, points $A$, $B$, $C$ and $D$ are on the same line and are the centers of four tangent circles at the same point. Segment $AB$ measures $8$ and segment $CD$ measures $4$. The circumferences woth centers $A$ and $C$ are of equal size. We know that the sum of the areas of the two medium circles is equivalent to the sum of the areas of the small and large circles. What is the length of segment $AD$?

We say that a positive integer is Noble when: it is composite, it is not divisible by any prime number greater than $20$ and it is not divisible by any perfect cube greater than $1$. How many different Noble numbers are there?

In a $2\times 2$ Domino game, each tile is square and divided into four spaces, as shown in the figure. In each box there is a number of points that varies from $0$ points (empty) to $6$ points. Two $2\times 2$ Domino tiles are equal if it is possible to rotate one of the two tiles until the other is obtained. In a $2\times 2$ Domino pack, what is the maximum number of different tiles that can be such that on each tile at least two squares have the same number of points?