Problems 6-10

Problem #6   $A$, $B$ and $C$ are building a wall. It takes $A$ and $B$ six days to build one-third of the wall. Then it takes $B$ and $C$ two days to build one-fourth of the remaining part. Finally it takes $A$ and $C$ another fifteen days to complete the wall. If $A$, $B$ and $C$ all works at the same time, how long does it take to build the wall?

Problem #7   Find all positive integers $m$, $n$, where $n$ is odd, that satisfy $\frac{1}{m}+\frac{4}{n}=\frac{1}{12}$.

Problem #8   Triangle $ABC$ has an area of $1$ square unit. $D$, $E$ are points on $AB$, while $F$, $G$ are points on $AC$ such that $AD=DE=EB$ and $AF=FG=GC$. Find the area of the quadrilateral bounded by $CD$, $CE$, $BF$ and $BG$.

Problem #9   A tutorial class has $5$ students. The teacher wants to rearrange the seats of students. If the chair and tables cannot be moved, and none of the students can be arranged to their original seats, then how many rearrangements are possible?

Problem #10   For $a,b,c>0$. Prove that $\frac{a}{(b+c{)}^4}+\frac{b}{(c+a{)}^4}+\frac{c}{(a+b{)}^4}\ge \frac{3}{2(a+b)(b+c)(c+a)}$.