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Explore the Back-Story(a) Find the positive integers $m, n$ such that $\frac{1}{m}+\frac{1}{n}=\frac{3}{17}$.

(b) Find the positive integers $m, n, p$ such that $\frac{1}{m}+\frac{1}{n}+\frac{1}{p}=\frac{3}{17}$.

Find the largest positive integer $n$ such that $3^n$ divides the $999$ digit number $9999 \ldots .99$.

Inside a square of area $36 \mathrm{~cm}^2$, there are shaded regions as shown. The ratio of the shaded area to the unshaded area is $3: 1$. What is the value of $a+b+c+d$ where $a, b, c, d$ are the lengths of the bases of the shaded regions ? Further, if three of $a, b, c, d$ are equal integers and one different, then find them.

Let the six faces of a cube be numbered $1,2,3,4,5,6$ in such a way that the 3 pairs $(1,6),(2,5),(3,4)$ lie on opposite faces of the cube. At each vertex of the cube, the product of the three numbers on the three faces containing the vertex is written. What is the sum of all the eight numbers written at the eight vertices of the cube?

Given a $2 \times 4$ rectangle with eight cells, find the total number of ways (frames) in which you can shade $75 \%$ of the cells. Few such frames are given below.

A square is divided into 5 identical rectangles as in the figure. Find the sum of the angles $\angle \mathrm{GBH}$, $\angle \mathrm{GCH}, \angle \mathrm{GDH}, \angle \mathrm{GEH}, \angle \mathrm{GFH}$. Given a valid proof for your answer.

Around a circle five positive integers $a, b, c, d, e$ are written in such a way that the sum of no three or no two adjacent integers is divisible by three. How many of these $a, b, c, d, e$ are divisible by three ? Please given proper proof for your answer.

Let $A B C D$ be a square with the length of side equal to $12 \mathrm{~cm}$. Points $P, Q, R$ are respectively the midpoints of side $B C, C D$ and $D A$ respectively (see figure). Find the area of the shaded region in square $\mathrm{cm}$. Given valid explanation for your steps.

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