[b]p1.[/b] An $8$-inch by $11$-inch sheet of paper is laid flat so that the top and bottom edges are $8$ inches long. The paper is then folded so that the top left corner touches the right edge. What is the minimum possible length of the fold?


[b]p2.[/b] Triangle $ABC$ is equilateral, with $AB = 6$. There are points $D$, $E$ on segment AB (in the order $A$, $D$, $E$, $B$), points $F$, $G$ on segment $BC$ (in the order $B$, $F$, $G$, $C$), and points $H$, $I$ on segment $CA$ (in the order $C$, $H$, $I$, $A$) such that $DE = F G = HI = 2$. Considering all such configurations of $D$, $E$, $F$, $G$, $H$, $I$, let $A_1$ be the maximum possible area of (possibly degenerate) hexagon $DEF GHI$ and let $A_2$ be the minimum possible area. Find $A_1 - A_2$.


[b]p3.[/b] Find $$\tan \frac{\pi}{7} \tan \frac{2\pi}{7} \tan \frac{3\pi}{7}$$


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To solve the problem, we need to find the minimum possible length of the fold when an 8-inch by 11-inch sheet of paper is folded such that the top left corner touches the right edge.

1. **Define the coordinates of the paper:**
   - Let the bottom-left corner be \( A(0, 0) \).
   - The bottom-right corner is \( B(8, 0) \).
   - The top-right corner is \( C(8, 11) \).
   - The top-left corner is \( D(0, 11) \).

2. **Determine the coordinates of the point where the top left corner touches the right edge:**
   - Let this point be \( E(8, y) \) where \( 0 \leq y \leq 11 \).

3. **Calculate the distance \( AE \):**
   - The distance \( AE \) is the length of the fold.
   - Using the distance formula, \( AE = \sqrt{(8-0)^2 + (y-11)^2} = \sqrt{64 + (y-11)^2} \).

4. **Minimize the length of the fold:**
   - To minimize \( AE \), we need to minimize \( \sqrt{64 + (y-11)^2} \).
   - This is equivalent to minimizing the expression inside the square root, \( 64 + (y-11)^2 \).

5. **Find the value of \( y \) that minimizes \( 64 + (y-11)^2 \):**
   - The expression \( 64 + (y-11)^2 \) is minimized when \( (y-11)^2 \) is minimized.
   - Since \( (y-11)^2 \) is a square term, it is minimized when \( y = 11 \).

6. **Calculate the minimum length of the fold:**
   - When \( y = 11 \), the length of the fold \( AE = \sqrt{64 + (11-11)^2} = \sqrt{64} = 8 \).

Therefore, the minimum possible length of the fold is \( 8 \) inches.

The final answer is \( \boxed{8} \).