Prove that there are no positive integers $x, y$ such that: $(x + 1)^2 + (x + 2)^2 +...+ (x + 9)^2 = y^2$

All diagonals are plotted in a $2017$-sided convex polygon. A line $\ell$ intersects said polygon but does not pass through any of its vertices. Show that the line $\ell$ intersects an even number of diagonals of said polygon.

Let $A, B, C, D$ be four different points on a line $\ell$, such that $AB = BC = CD$. In one of the semiplanes determined by the line $\ell$, the points $P$ and $Q$ are chosen in such a way that the triangle $CPQ$ is equilateral with its vertices named clockwise. Let $M$ and $N$ be two points on the plane such that the triangles $MAP$ and $NQD$ are equilateral (the vertices are also named clockwise). Find the measure of the angle $\angle MBN$.

In the parallelogram $ABCD$, a line through $C$ intersects the diagonal $BD$ at $E$ and $AB$ at $F$. If $F$ is the midpoint of $AB$ and the area of $\vartriangle BEC$ is $100$, find the area of the quadrilateral $AFED$.

Determine the number of positive integers $N = \overline{abcd}$, with $a, b, c, d$ nonzero digits, which satisfy $(2a -1) (2b -1) (2c- 1) (2d - 1) = 2abcd -1$.

A positive integer $n$ is "olympic" if there are $n$ non-negative integers $x_1, x_2, ..., x_n$ that satisfy that: $\bullet$ There is at least one positive integer $j$: $1 \le j \le n$ such that $x_j \ne 0$. $\bullet$ For any way of choosing $n$ numbers $c_1, c_2, ..., c_n$ from the set $\{-1, 0, 1\}$, where not all $c_i$ are equal to zero, it is true that the sum $c_1x_1 + c_2x_2 +... + c_nx_n$ is not divisible by $n^3$. Find the largest positive "olympic" integer.