Let $a_1,a_2,a_3, \ldots$ be an infinite sequence of nonnegative real numbers such that for all positive integers $k$ the following conditions hold: $i)$ $a_k-2a_{k+1}+a_{k+2} \geq 0$; $ii)$ $\sum_{j=1}^{k} a_j \leq 1$. Prove that for all positive integer $k$ holds: $0 \leq a_k - a_{k+1} < \frac{2}{k^2}$

Determine all positive integers $A= \overline{a_n a_{n-1} \ldots a_1 a_0}$ such that not all of its digits are equal and no digit is $0$, and $A$ divides all numbers of the following form: $A_1 = \overline{a_0 a_n a_{n-1} \ldots a_2 a_1}, A_2 = \overline{a_1 a_0 a_{n} \ldots a_3 a_2}, \ldots ,$ $ A_{n-1} = \overline{a_{n-2} a_{n-3} \ldots a_0 a_n a_{n-1}}, A_n = \overline{a_{n-1} a_{n-2} \ldots a_1 a_0 a_n}$.

Cyclic quadrilateral $ABCD$ is inscribed in circle $k$ with center $O$. The angle bisector of $ABD$ intersects $AD$ and $k$ in $K,M$ respectively, and the angle bisector of $CBD$ intersects $CD$ and $k$ in $L,N$ respectively. If $KL\parallel MN$ prove that the circumcircle of triangle $MON$ bisects segment $BD$.