MathDB

p1. Find the shortest path from point

A

to point

B

that does not pass through the interior of the circular region. https://cdn.artofproblemsolving.com/attachments/4/4/192e3d15a2af927e060235f83fe3ca5ed805d0.jpg
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 harder P1, P2, P3, P5, harder P6.

2011 Chile Classification NMO Juniors

Subcontests

2011 Chile Classification / Qualifying NMO Juniors XXIII