p1. Find the shortest path from point $A$ to point $B$ that does not pass through the interior of the circular region. 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 Seniors harder P1, P2, P3, P5, harder P6.