p1. Find the shortest path from point $A$ to point $B$ that does not pass through the interior of the circular region.
[img]https://cdn.artofproblemsolving.com/attachments/4/4/192e3d15a2af927e060235f83fe3ca5ed805d0.jpg[/img]


p2. A giant, circular and perfectly flat pizza must be shared by $22$ people. Which is the least number of cuts to be made on the pizza so that each person can  have a piece of pizza? (the pieces are not necessarily the same shape or with the same area).


p3.Are there integers $n, m$ such that the equation $n\sqrt2 + m\sqrt3 = 2011$ will hold?


p4. An convex quadrilateral is drawn and the $4$ triangles obtained by $3$ of the $4$ vertices of the quadrilateral . If the area of the largest of these triangles is $1888$ and the area of the the smallest of these triangles is $123$, determine what is the greatest value that the area of the quadrilateral can have.


p5. Determine whether or not there are two digits other than $a, b$ such that the number $\overline{ab}$ is a multiple of the number $\overline{ba}$ (both written in decimal notation).


p6. On an infinite lagoon there are arranged lotus flowers numbered $f_1, f_2, f_3, ...$ .Above each of the first $25$ lotus flowers there is a little frog. The frog  jump out of a lotus flower according to the following rule: if a frog is on the $f_n$ , he can jump to the $f_{n + 1}$   or at $f_{n + 30}$ (as it likes) .  Show that frogs can jump so that each lotus flower is  visited by a frog exactly once.


PS. Juniors P1, P2, P3, P5, P6 were also proposed as [url=https://artofproblemsolving.com/community/c4h2693399p23385819]Seniors[/url] harder P1, P2, P3, P5, harder P6.
### Problem 1:
1. To find the shortest path from point \( A \) to point \( B \) that does not pass through the interior of the circular region, we need to consider the geometry of the circle and the points \( A \) and \( B \).
2. Draw tangents from points \( A \) and \( B \) to the circle. Let the points where these tangents touch the circle be \( P \) and \( Q \) respectively.
3. The shortest path that avoids the interior of the circle will be the path \( A \to P \to Q \to B \), where \( P \) and \( Q \) are the points of tangency.
4. The path \( A \to P \) and \( B \to Q \) are straight lines, and the path \( P \to Q \) is along the circumference of the circle.

The final answer is \( \boxed{\text{True}} \).