For what integers $ n\ge 3$ is it possible to accommodate, in some order, the numbers $ 1,2,\cdots, n$ in a circular form such that every number divides the sum of the next two numbers, in a clockwise direction?

Let $ ABC$ be an acute triangle, and let $ D$ and $ E$ be the feet of the altitudes drawn from vertexes $ A$ and $ B$, respectively. Show that if, \[ Area[BDE]\le Area[DEA]\le Area[EAB]\le Area[ABD]\] then, the triangle is isosceles.

For every integer $ a>1$ an infinite list of integers is constructed $ L(a)$, as follows: $ a$ is the first number in the list $ L(a)$. Given a number $ b$ in $ L(a)$, the next number in the list is $ b+c$, where $ c$ is the largest integer that divides $ b$ and is smaller than $ b$. Find all the integers $ a>1$ such that $ 2002$ is in the list $ L(a)$.

Let $ ABC$ be a triangle, $ D$ be the midpoint of $ BC$, $ E$ be a point on segment $ AC$ such that $ BE=2AD$ and $ F$ is the intersection point of $ AD$ with $ BE$. If $ \angle DAC=60^{\circ}$, find the measure of the angle $ FEA$.

Find a set of infinite positive integers $ S$ such that for every $ n\ge 1$ and whichever $ n$ distinct elements $ x_1,x_2,\cdots, x_n$ of S, the number $ x_1+x_2+\cdots +x_n$ is not a perfect square.

A path from $ (0,0)$ to $ (n,n)$ on the lattice is made up of unit moves upward or rightward. It is balanced if the sum of the x-coordinates of its $ 2n+1$ vertices equals the sum of their y-coordinates. Show that a balanced path divides the square with vertices $ (0,0)$, $ (n,0)$, $ (n,n)$, $ (0,n)$ into two parts with equal area.