Find the smallest positive integer $n$ that satisfies that for any $n$ different integers, the product of all the positive differences of these numbers is divisible by $2014$.

Let $x_1$, $x_2$,$x_3$, $y_1$, $y_2$, and $y_3 $ be positive real numbers, such that $x_1 + y_2 = x_2 + y_3 = x_3 + y_1 =1$. Prove that $$ x_1y_1 + x_2y_2 + x_3y_3 <1$$

Let $AB$ be a triangle and $\Gamma$ the excircle, relative to the vertex $A$, with center $D$. The circle $\Gamma$ is tangent to the lines $AB$ and $AC$ at $E$ and $F$, respectively. Let $P$ and $Q$ be the intersections of $EF$ with $BD$ and $CD$, respectively. If $O$ is the point of intersection of $BQ$ and $CP$, show that the distance from $O$ to the line $BC$ is equal to the radius of the inscribed circle in the triangle $ABC$.

Let $ABCD$ be a square and let $M$ be the midpoint of $BC$. Let $C ^ \prime$ be the reflection of $C$ wrt to $DM$. The parallel to $AB$ passing through $C ^ \prime$ intersects $AD$ at $R$ and $BC$ at $S$. Show that $$\frac {RC ^ \prime} {C ^\prime S} = \frac {3} {2}$$

In a school there are $n$ classes and $n$ students. The students are enrolled in classes, such that no two of them have exactly the same classes. Prove that we can close a class in a such way that there still are no two of them which have exactly the same classes.