Question bank USAMO 2001

Each of eight boxes contains six balls.
Each ball has been colored with one of $n$ colors,
such that no two balls in the same box are the same color,
and no two colors occur together in more than one box.
Find with proof the smallest possible $n$.

Let $ABC$ be a triangle and let $\omega$ be its incircle.
Denote by $D_1$ and $E_1$ the points where $\omega$
is tangent to sides $BC$ and $AC$, respectively.
Denote by $D_2$ and $E_2$ the points on sides $BC$ and $AC$,
respectively, such that $CD_2=BD_1$ and $CE_2=AE_1$,
and denote by $P$ the point of intersection of segments $AD_2$ and $BE_2$.
Circle $\omega$ intersects segment $AD_2$ at two points,
the closer of which to the vertex $A$ is denoted by $Q$.
Prove that $AQ=D_2P$.

Let $a, b, c$ be nonnegative real numbers
such that $a^2+b^2+c^2 + abc = 4$.
Show that \[ 0 \le ab + bc + ca - abc \le 2. \]

Let $ABC$ be a triangle and $P$ any point
such that $PA$, $PB$, $PC$ are the sides of an obtuse triangle,
with $PA$ the longest side.
Prove that $\angle BAC$ is acute.

Let $S\subseteq\mathbb Z$ satisfy:

(a) There exist $a,b\in S$ with $\gcd(a,b)=\gcd(a-2,b-2)=1$.

(b) If $x,y\in S$ (possibly equal), then $x^2-y\in S$.

Prove that $S=\mathbb Z$.

Each point in the plane is assigned a real number.
Suppose that for any nondegenerate triangle, the number at its incenter
is the arithmetic mean of the three numbers at its vertices.
Prove that all points in the plane were assigned the same number.