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Chile Classification NMO
2011 Chile Classification NMO Seniors
2011 Chile Classification NMO Seniors
Part of
Chile Classification NMO
Subcontests
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2011 Chile Classification / Qualifying NMO Seniors XXIII
p1. Find the shortest path from point
A
A
A
to point
B
B
B
that does not pass through the circular region. Calculate the length of such a path if
A
O
=
O
B
=
2
AO = OB = 2
A
O
=
OB
=
2
and the radius of the circular region is
1
1
1
. https://cdn.artofproblemsolving.com/attachments/4/4/192e3d15a2af927e060235f83fe3ca5ed805d0.jpg p2. A giant, circular and perfectly flat pizza must be shared by
211
211
211
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
n, m
n
,
m
such that the equation
n
2
+
m
3
=
2011
n\sqrt2 + m\sqrt3 = 2011
n
2
+
m
3
=
2011
will hold? p4. A swimmer is in the center of a circular pool. At the edge of the pool a dog waits for him to try to bite him. The dog may only run along the edge of the pool at a speed
4
4
4
times greater than the speed with which the swimmer advances in the water. Determine if it is possible or not that the swimmer can leave the pool without being hit by the dog. p5. Determine whether or not there are two digits other than
a
,
b
a, b
a
,
b
such that the number
a
b
‾
\overline{ab}
ab
is a multiple of the number
b
a
‾
\overline{ba}
ba
(both written in decimal notation). p6. On an infinite lagoon there are arranged lotus flowers numbered
f
1
,
f
2
,
f
3
,
.
.
.
f_1, f_2, f_3,...
f
1
,
f
2
,
f
3
,
...
On top of each of the first
k
k
k
lotus flowers there is a little frog. The little frogs jump from one flower to another following the following rule: if a little frog is on the flower
f
n
f_n
f
n
it can jump (as it likes) to the flower
f
n
+
1
f_{n + 1}
f
n
+
1
or to the flower
f
n
+
m
f_{n + m}
f
n
+
m
where
m
>
1
m> 1
m
>
1
is a fixed integer number and is the same for all frogs. Show that if
k
≤
m
k \le m
k
≤
m
then the frogs can do jumps so that each flower is visited by a frog exactly once. Show furthermore, this is not possible when
k
>
m
k> m
k
>
m
.PS. Seniors P1, P2, P3, P5, P6 were also proposed as [url=https://artofproblemsolving.com/community/c4h2690186p23348597]Juniors easier P1, P2, P3, P5m easier of P6.