Problem Corner

Welcome to the problem corner of Mathematics Club.

Newest Problems

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 $\frac{a}{(b+c{)}^4}+\frac{b}{(c+a{)}^4}+\frac{c}{(a+b{)}^4}\ge \frac{3}{2(a+b)(b+c)(c+a)}$.

Rules

1. Both members and non-members can submit solutions.

2. You can submit the solution of ONE OR MORE questions. Each problem can only be attempted once.

3. Please use A4-sized paper to submit solutions. Remember to write down your name, class and class number.

4. All answers MUST be submitted with working steps. Solutions without working steps will be disqualified.

5. Students who submit one or more correct solutions are eligible to win prizes. Prizes are determined by lucky draw.

6. All qualified submissions (including wrong answers) are counted as participation in Mathematics Society.

How to submit solutions

1. You can send solutions to Mr. Leung by email ([email protected]) or by hand.

2. To obtain a soft copy of your solution, use a suitable app (e.g. Office Lens) to scan your paper.