Let the three roots of $x^3+ax+1=0$ be $\alpha,\beta,\gamma$ where $a$ is a positive real number. Let the three roots of $x^3+bx^2+cx-1=0$ be $\frac{\alpha}{\beta},\frac{\beta}{\gamma},\frac{\gamma}{\alpha}$. Find the minimum value of $\dfrac{|b|+|c|}{a}$.

(a) Let the incenter of $\triangle ABC$ be $I$. We connect $I$ other $3$ vertices and divide $\triangle ABC$ into $3$ small triangles which has area $2,3$ and $4$. Find the area of the inscribed circle of $\triangle ABC$. (b) Let $ABCD$ be a parallelogram. Point $E,F$ is on $AB,BC$ respectively. If $[AED]=7,[EBF]=3,[CDF]=6$, then find $[DEF].$ (Here $[XYZ]$ denotes the area of $XYZ$)

Let a board game has $10$ cards: $3$ skull cards, $5$ coin cards and $2$ blank cards. We put these $10$ cards downward and shuffle them and take cards one by one from the top. Once $3$ skull cards or coin cards appears we stop. What is the possibility of it stops because there appears $3$ skull cards?

Let $n$ be a positive integer. All numbers $m$ which are coprime to $n$ all satisfy $m^6\equiv 1\pmod n$. Find the maximum possible value of $n$.

$\triangle ABC$, $\angle A=23^{\circ},\angle B=46^{\circ}$. Let $\Gamma$ be a circle with center $C$, radius $AC$. Let the external angle bisector of $\angle B$ intersects $\Gamma$ at $M,N$. Find $\angle MAN$.

Find all positive integers $A,B$ satisfying the following properties: (i) $A$ and $B$ has same digit in decimal. (ii) $2\cdot A\cdot B=\overline{AB}$ (Here $\cdot$ denotes multiplication, $\overline{AB}$ denotes we write $A$ and $B$ in turn. For example, if $A=12,B=34$, then $\overline{AB}=1234$)

Let $n$ be a fixed positive integer. We have a $n\times n$ chessboard. We call a pair of cells good if they share a common vertex (May be common edge or common vertex). How many good pairs are there on this chessboard?