Find all positive integers $\overline{xyz}$ ($x$, $y$ and $z$ are digits) such that $\overline{xyz} = x+y+z+xy+yz+zx+xyz$

Let $A$ be a set $A=\{1,2,3,...,2017\}$. Subset $S$ of set $A$ is good if for all $x\in A$ sum of remaining elements of set $S$ has same last digit as $x$. Prove that good subset with $405$ elements is not possible.

Let $ABC$ be a triangle such that $\angle ABC = 90 ^{\circ}$. Let $I$ be an incenter of $ABC$ and let $F$, $D$ and $E$ be points where incircle touches sides $AB$, $BC$ and $AC$, respectively. If lines $CI$ and $EF$ intersect at point $M$ and if $DM$ and $AB$ intersect in $N$, prove that $AI=ND$

In each cell of $5 \times 5$ table there is one number from $1$ to $5$ such that every number occurs exactly once in every row and in every column. Number in one column is good positioned if following holds: - In every row, every number which is left from good positoned number is smaller than him, and every number which is right to him is greater than him, or vice versa. - In every column, every number which is above from good positoned number is smaller than him, and every number which is below to him is greater than him, or vice versa. What is maximal number of good positioned numbers that can occur in this table?