A cyclic quadrilateral $ABCD$ is given. Point $K_1, K_2$ lie on the segment $AB$, points $L_1, L_2$ on the segment $BC$, points $M_1, M_2$ on the segment $CD$ and points $N_1, N_2$ on the segment $DA$. Moreover, points $K_1, K_2, L_1, L_2, M_1, M_2, N_1, N_2$ lie on a circle $\omega$ in that order. Denote by $a, b, c, d$ the lengths of the arcs $N_2K_1, K_2L_1, L_2M_1, M _2N_1$ of the circle $\omega$ not containing points $K_2, L_2, M_2, N_2$, respectively. Prove that \begin{align*} a+c=b+d. \end{align*}

Determine all nonnegative integers $x, y$ satisfying the equation \begin{align*} \sqrt{xy}=\sqrt{x+y}+\sqrt{x}+\sqrt{y}. \end{align*}

Let $f(t)=t^3+t$. Decide if there exist rational numbers $x, y$ and positive integers $m, n$ such that $xy=3$ and: \begin{align*} \underbrace{f(f(\ldots f(f}_{m \ times}(x))\ldots)) = \underbrace{f(f(\ldots f(f}_{n \ times}(y))\ldots)). \end{align*}

Let $a_1, a_2, \ldots, a_n$ ($n\ge 3$) be positive integers such that $gcd(a_1, a_2, \ldots, a_n)=1$ and for each $i\in \lbrace 1,2,\ldots, n \rbrace$ we have $a_i|a_1+a_2+\ldots+a_n$. Prove that $a_1a_2\ldots a_n | (a_1+a_2+\ldots+a_n)^{n-2}$.

Let $b_0, b_1, b_2, \ldots$ be a sequence of pairwise distinct nonnegative integers such that $b_0=0$ and $b_n<2n$ for all positive integers $n$. Prove that for each nonnegative integer $m$ there exist nonnegative integers $k, \ell$ such that \begin{align*} b_k+b_{\ell}=m. \end{align*}

Let $X$ be a point lying in the interior of the acute triangle $ABC$ such that \begin{align*} \sphericalangle BAX = 2\sphericalangle XBA \ \ \ \ \hbox{and} \ \ \ \ \sphericalangle XAC = 2\sphericalangle ACX. \end{align*}Denote by $M$ the midpoint of the arc $BC$ of the circumcircle $(ABC)$ containing $A$. Prove that $XM=XA$.