IL-GROSSMAN 2001

Find all real solutions of the system $$\begin{cases} x_1 +x_2 +...+x_{2000} = 2000 x_1^4 +x_2^4 +...+x_{2000}^4= x_1^3 +x_2^3 +...+x_{2000}^3\end{cases}$$

If $x_1,x_2,...,x_{2001}$ are real numbers with $0 \le x_n \le 1$ for $n = 1,2,...,2001$ , find the maximum value of $$\left(\frac{1}{2001}\sum_{n=1}^{2001}x_n^2\right)-\left(\frac{1}{2001}\sum_{n=1}^{2001}x_n\right)^2$$ Where is this maximum attained?

We are given $2001$ lines in the plane, no two of which are parallel and no three of which are concurrent. These lines partition the plane into regions (not necessarily finite) bounded by segments of these lines. These segments are called sides, and the collection of the regions is called a map. Intersection points of the lines are called vertices. Two regions are neighborsif they share a side, and two vertices are neighbors if they lie on the same side. A legal coloring of the regions (resp. vertices) is a coloring in which each region (resp. vertex) receives one color, such that any two neighboring regions (vertices) have different colors. (a) What is the minimum number of colors required for a legal coloring of the regions? (b) What is the minimum number of colors required for a legal coloring of the vertices?

The lengths of the sides of triangle $ABC$ are $4,5,6$ . For any point $D$ on one of the sides, draw the perpendiculars $DP, DQ$ on the other two sides. What is the minimum value of $PQ$ ?