A convex quadrilateral $ABCD$ is given in which the bisectors of the interior angles $\angle ABC$ and $\angle ADC$ have a common point on the diagonal $AC$. Prove that the bisectors of the interior angles $\angle BAD$ and $\angle BCD$ have a common point on the diagonal $BD$.

One positive integer is written in each $1 \times 1$ square of the $m \times n$ board. The following operations are allowed : (1) In an arbitrarily selected row of the board, all numbers should be reduced by $1$. (2) In an arbitrarily selected column of the board, double all the numbers. Is it always possible, after a final number of steps, for all the numbers written on the board to be equal to $-1$? (Explain the answer.)

For given integers $n>0$ and $k> 1$, let $F_{n,k}(x,y)=x!+n^k+n+1-y^k$. Prove that there are only finite couples $(a,b)$ of positive integers such that $F_{n,k}(a,b)=0$