MathDB

Some time ago there was a war across the world. In the plane

n

lines are moving, with the regions contained by the lines being the territories of the countries at war. Each line moves parallel to itself with constant speed (each with its own speed), and no line can reverse its direction. Some of the original countries disappeared (a country disappears iff its area is converted to zero) and within the course of the time, other countries appeared.
After some time, the presidents of the existing countries made a treaty to end the war, created the United Nations, and all borders ceased movement. The UN then counted the total numbers of sovereign states that were destroyed and the existing ones, obtaining a total of

k

.
Prove that

k\leq \frac{n^3+5n}{6}+1

. Is is possible to have equality?

Problem Statement