Let $p$ a prime number and $r$ the remainder of the division of $p$ by $210$. It is known that $r$ is a composite number and can be written as the sum of two non-zero perfect squares. Find all primes less than $2018$ that satisfy these conditions.

There are $n$ knights numbered $1$ to $n$ and a round table with $n$ chairs. The first knight chooses his chair, and from him, the knight number $k+1$ sits $ k$ places to the right of knight number $k$ , for all $1 \le k\le n-1$ (occupied and empty seats are counted). In particular, the second knight sits next to the first. Find all values of $n$ such that the $n$ gentlemen occupy the $n$ chairs following the described procedure.

You have a $7\times 7$ board divided into $49$ boxes. Mateo places a coin in a box. a) Prove that Mateo can place the coin so that it is impossible for Emi to completely cover the $48$ remaining squares, without gaps or overlaps, using $15$ $3\times1$ rectangles and a cubit of three squares, like those in the figure. b) Prove that no matter which square Mateo places the coin in, Emi will always be able to cover the 48 remaining squares using $14$ $3\times1$ rectangles and two cubits of three squares.

There is a $50\times 50$ grid board.. Carlos is going to write a number in each box with the following procedure. He first chooses $100$ distinct numbers that we denote $f_1,f_2,f_3,…,f_{50},c_1,c_2,c_3,…,c_{50}$ among which there are exactly $50$ that they are rational. Then he writes in each box ($i,j)$ the number $f_i \cdot c_j$ (the multiplication of $f_i$ by $c_j$). Determine the maximum number of rational numbers that the squares on the board can contain.

In the plane you have $2018$ points between which there are not three on the same line. These points are colored with $30$ colors so that no two colors have the same number of points. All triangles are formed with their three vertices of different colors. Determine the number of points for each of the $30$ colors so that the total number of triangles with the three vertices of different colors is as large as possible.

Let $ABCD$ be a parallelogram. An interior circle of the $ABCD$ is tangent to the lines $AB$ and $AD$ and intersects the diagonal $BD$ at $E$ and $F$. Prove that exists a circle that passes through $E$ and $F$ and is tangent to the lines $CB$ and $CD$.