The diagonals $ AC$ and $ BD$ of a convex quadrilateral $ ABCD$ intersect at point $ M$. The bisector of $ \angle ACD$ meets the ray $ BA$ at $ K$. Given that $ MA \cdot MC \plus{}MA \cdot CD \equal{} MB \cdot MD$, prove that $ \angle BKC \equal{} \angle CDB$.

$ABCD$ is a convex quadrilateral. $P$ is the intersection of external bisectors of $\angle DAC$ and $\angle DBC$. Prove that $\angle APD = \angle BPC$ if and only if $AD+AC=BC+BD$

The diagonals $ AC$ and $ BD$ of a convex quadrilateral $ ABCD$ intersect at point $ M$. The bisector of $ \angle ACD$ meets the ray $ BA$ at $ K$. Given that $ MA \cdot MC \plus{}MA \cdot CD \equal{} MB \cdot MD$, prove that $ \angle BKC \equal{} \angle CDB$.

The diagonals $ AC$ and $ BD$ of a convex quadrilateral $ ABCD$ intersect at point $ M$. The bisector of $ \angle ACD$ meets the ray $ BA$ at $ K$. Given that $ MA \cdot MC \plus{}MA \cdot CD \equal{} MB \cdot MD$, prove that $ \angle BKC \equal{} \angle CDB$.

Robot "Mag-o-matic" manipulates $101$ glasses, displaced in a row whose positions are numbered from $1$ to $101$. In each glass you can find a ball or not. Mag-o-matic only accepts elementary instructions of the form $(a;b,c)$, which it interprets as "consider the glass in position $a$: if it contains a ball, then switch the glasses in positions $b$ and $c$ (together with their own content), otherwise move on to the following instruction" (it means that $a,\,b,\,c$ are integers between $1$ and $101$, with $b$ and $c$ different from each other but not necessarily different from $a$). A $\emph{programme}$ is a finite sequence of elementary instructions, assigned at the beginning, that Mag-o-matic does one by one. A subset $S\subseteq \{0,\,1,\,2,\dots,\,101\}$ is said to be $\emph{identifiable}$ if there exists a programme which, starting from any initial configuration, produces a final configuration in which the glass in position $1$ contains a ball if and only if the number of glasses containing a ball is an element of $S$. (a) Prove that the subset $S$ of the odd numbers is identifiable. (b) Determine all subsets $S$ that are identifiable.