Each one of $2001$ children chooses a positive integer and writes down his number and names of some of other $2000$ children to his notebook. Let $A_c$ be the sum of the numbers chosen by the children who appeared in the notebook of the child $c$. Let $B_c$ be the sum of the numbers chosen by the children who wrote the name of the child $c$ into their notebooks. The number $N_c = A_c - B_c$ is assigned to the child $c$. Determine whether all of the numbers assigned to the children could be positive.

A circle touches to diameter $AB$ of a unit circle with center $O$ at $T$ where $OT>1$. These circles intersect at two different points $C$ and $D$. The circle through $O$, $D$, and $C$ meet the line $AB$ at $P$ different from $O$. Show that \[|PA|\cdot |PB| = \dfrac {|PT|^2}{|OT|^2}.\]

For all integers $x,y,z$, let \[S(x,y,z) = (xy - xz, yz-yx, zx - zy).\] Prove that for all integers $a$, $b$ and $c$ with $abc>1$, and for every integer $n\geq n_0$, there exists integers $n_0$ and $k$ with $0<k\leq abc$ such that \[S^{n+k}(a,b,c) \equiv S^n(a,b,c) \pmod {abc}.\] ($S^1 = S$ and for every integer $m\geq 1$, $S^{m+1} = S \circ S^m.$ $(u_1, u_2, u_3) \equiv (v_1, v_2, v_3) \pmod M \Longleftrightarrow u_i \equiv v_i \pmod M (i=1,2,3).$)

Find all ordered pairs of integers $(x,y)$ such that $5^x = 1 + 4y + y^4$.

Let $H$ be the intersection of the altitudes of an acute triangle $ABC$ and $D$ be the midpoint of $[AC]$. Show that $DH$ passes through one of the intersection point of the circumcircle of $ABC$ and the circle with diameter $[BH]$.

Show that there is no continuous function $f:\mathbb{R}\rightarrow \mathbb{R}$ such that for every real number $x$ \[f(x-f(x)) = \dfrac x2.\]