We define the set $M = \{1^2,2^2,3^2,..., 99^2, 100^2\}$. a) What is the smallest positive integer that divides exactly two elements of $M$? b) What is the largest positive integer that divides exactly two elements of $M$?

For each positive real number $x$, let $f(x)=\frac{x}{1+x}$ . Prove that if $a$, $b,$ $c$ are the sidelengths of a triangle, then $f(a)$, $f(b),$ $f(c)$ are sidelengths of a triangle.

Prove that, for every integer $n \ge 2$, it is possible to divide a regular hexagon into $n$ quadrilaterals such that any two of them are similar. Clarification: Two quadrilaterals are similar if they have their corresponding sides proportional and their corresponding angles are equal, that is, the quadrilaterals $ABCD$ and $EFGH$ are similar if $\frac{AB}{EF}= \frac{BC}{FG}= \frac{CD}{GH} = \frac{DA}{HE}$, $\angle ABC = \angle EFG$, $\angle BCD = \angle FGH$, $\angle CDA = \angle GHE$ and $\angle DAB = \angle HEF$.

Let $ABC$ be an acute scalene triangle and $K$ be a point inside it that belongs to the bisector of the angle $\angle ABC$. Let$ P$ be the point where the line $AK$ intersects the line perpendicular to $AB$ that passes through $B$, and let $Q$ be the point where the line $CK$ intersects the line perpendicular to $CB$ that passes through $B$. Let $L$ be the foot of the perpendicular drawn from $K$ on the line $AC$. Prove that if $P Q$ is perpendicular to $BL$, then $K$ is the incenter of $ABC$.