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Chile Classification NMO
2017 Chile Classification NMO Seniors
2017 Chile Classification NMO Seniors
Part of
Chile Classification NMO
Subcontests
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2017 Chile Classification / Qualifying NMO Seniors XXIX
p1. Inside a rectangle a point
P
P
P
is marked. Segments are drawn that join the vertices with
P
P
P
. In an alternate way, the sectors that are formed are colored. Show that the sum of the areas of the painted sectors is equal to the sum of the unpainted sectors. p2. Find the maximum number of different paths that can be built on a pool table to join two balls on the
n
n
n
-cushion table. No ball is touching a band. A band is defined when a ball bounces off one side of the table, with the angle of incidence equal to the angle of departure. https://cdn.artofproblemsolving.com/attachments/0/4/cc36d945d7a14c544ba0c6662861f1526e0f5d.jpg p3. Using only two different digits
2
2
2
and
d
d
d
, the following
90
90
90
-digit number is formed:
m
=
2
d
22
d
222
d
.
.
.
m= 2d22d222d...
m
=
2
d
22
d
222
d
...
If
m
m
m
is a multiple of
9
9
9
, determine all possible values of the digit
d
d
d
. p4. Calculate all the solutions
x
,
y
,
z
x,y,z
x
,
y
,
z
in the positive real numbers of the following system:
x
(
6
−
y
)
=
9
,
y
(
6
−
z
)
=
9
,
z
(
6
−
x
)
=
9
x(6- y) = 9\,\, ,\,\,y(6-z) = 9\,\, ,\,\ z(6-x) = 9
x
(
6
−
y
)
=
9
,
y
(
6
−
z
)
=
9
,
z
(
6
−
x
)
=
9
PS. Seniors p1 was also [url=https://artofproblemsolving.com/community/c4h2690911p23356729]Juniors p1.