In the following figure, it is desired to go from point \( A \) to point \( B \) by walking only along the lines of the figure up and to the right. How many different paths can we take?

Let \( L \), \( M \) and \( N \) be the midpoints on the sides \( BC \), \( AC \) and \( AB\) of a triangle \( ABC \). Points \( D \), \( E \) and \( F \) are taken on the circle circumscribed to the triangle \( LMN \) so that the segments \( LD \), \( ME \) and \( NF \) are diameters of said circumference. Prove that the area of the hexagon \( LENDMF \) is equal to half the area of the triangle \( ABC \)

Prove that for every natural number \( n>2 \) there exists an integer \( k \) that can be written as the sum of \( i \) positive perfect squares, for every \( i \) between \( 2 \) and \( n \).

Given a positive integer \( n \), we denote by \( P(n) \) the result of multiplying all the digits of \( n \). Find a number \( m \) with ten digits, none of them zero, with the following property: $$P\left(m+P(m)\right)= P (m)$$

Determine the values that \(n\) can take so that the equation in \( x \) $$ x^4-(3n+2)x^2+n^2=0$$has four different real roots \( x_1\), \(x_2\), \(x_3\) and \(x_4\) in arithmetic progression. That is, they satisfy that $$x_4-x_3=x_3-x_2=x_2-x_1$$

Let \( M \) be the midpoint of side \( BC \) of a scalene triangle \( ABC \). The circle passing through \( A \), \( B \) and \( M \) intersects side \( AC \) again at \( D \). The circle passing through \( A \), \( C \) and \( M \) cuts side \( AB \) again at \( E \). Let \( O \) be the circumcenter of triangle \( ADE \). Prove that \( OB=OC \).