(a) Find the minimal distance between the points of the graph of the function $y=\ln x$ from the line $y=x$. (b) Find the minimal distance between two points, one of the point is in the graph of the function $y=e^x$ and the other point in the graph of the function $y=ln x$.

Find all the values that can take the last digit of a "perfect" even number. (The natural number $n$ is called "perfect" if the sum of all its natural divisors is equal twice the number itself.For example: the number $6$ is perfect ,because $1+2+3+6=2\cdot6$).

In a convex quadrilateral $ABCD$ ,$\angle ABC$ and $\angle BCD$ are $\geq 120^o$. Prove that $|AC|$ + $|BD| \geq |AB|+|BC|+|CD|$. (With $|XY|$ we understand the length of the segment $XY$).

The sequence $(a_{n})$ is defined by $a_1=1$ and $a_n=n(a_1+a_2+\cdots+a_{n-1})$ , $\forall n>1$. (a) Prove that for every even $n$, $a_{n}$ is divisible by $n!$. (b) Find all odd numbers $n$ for the which $a_{n}$ is divisible by $n!$.

The triangle $ABC$ acute with gravity center $M$ is such that $\angle AMB = 2 \angle ACB$. Prove that: (a) $AB^4=AC^4+BC^4-AC^2 \cdot BC^2,$ (b) $\angle ACB \geq 60^o$.