Is there $14$ consecutive positive integers such that each of these numbers is divisible by one of the prime numbers $p$ where $2\leq p \leq 11$.

The line passing through $B$ is perpendicular to the side $AC$ at $E$. This line meets the circumcircle of $\triangle ABC$ at $D$. The foot of the perpendicular from $D$ to the side $BC$ is $F$. If $O$ is the center of the circumcircle of $\triangle ABC$, prove that $BO$ is perpendicular to $EF$.

$x_1, x_2,\cdots,x_{n+1}$ are posive real numbers satisfying the equation $\frac{1}{1+x_1} + \frac{1}{1+x_2} + \cdots + \frac{1}{1+x_{n+1}} =1$ Prove that $x_1x_2 \cdots x_{n+1} \geq n^{n+1}$.

The feet of perpendiculars from the intersection point of the diagonals of cyclic quadrilateral $ABCD$ to the sides $AB,BC,CD,DA$ are $P,Q,R,S$, respectively. Prove $PQ+RS=QR+SP$.

There are $n$ boxes which is numbere from $1$ to $n$. The box with number $1$ is open, and the others are closed. There are $m$ identical balls ($m\geq n$). One of the balls is put into the open box, then we open the box with number $2$. Now, we put another ball to one of two open boxes, then we open the box with number $3$. Go on until the last box will be open. After that the remaining balls will be randomly put into the boxes. In how many ways this arrangement can be done?

A circle with radius $4$ and $251$ distinct points inside the circle are given. Show that it is possible to draw a circle with radius $1$ and containing at least $11$ of these points.