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Chile Classification NMO
2019 Chile Classification NMO Seniors
2019 Chile Classification NMO Seniors
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Chile Classification NMO
Subcontests
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2019 Chile Classification / Qualifying NMO Seniors XXXI
p1. The sequence of numbers
123456789101112131415...
123456789101112131415...
123456789101112131415...
is obtained by writing the positive integers in order, one after the other. What position is
2
2
2
in the first time does
2019
2019
2019
appear in succession? p2. Consider a square in the plane with vertices
(
a
1
,
b
1
)
,
(
a
2
,
b
2
)
,
(
a
3
,
b
3
)
,
(
a
4
,
b
4
)
(a_1, b_1), (a_2, b_2), (a_3, b_3), (a_4, b_4)
(
a
1
,
b
1
)
,
(
a
2
,
b
2
)
,
(
a
3
,
b
3
)
,
(
a
4
,
b
4
)
where
a
i
a_i
a
i
,
b
i
b_i
b
i
are integer numbers for each
i
=
1
,
.
.
.
,
4
i = 1, ..., 4
i
=
1
,
...
,
4
. Suppose the area of the square is a power of
3
3
3
. Prove that its sides are parallel to the axes. p3. Prove that for every integer
n
>
2
n> 2
n
>
2
, it is true that
1
n
+
1
+
1
n
+
2
+
.
.
.
+
1
2
n
<
5
6
\frac{1}{n + 1}+ \frac{1}{n + 2}+ ...+ \frac{1}{2n}< \frac56
n
+
1
1
+
n
+
2
1
+
...
+
2
n
1
<
6
5
p4.
A
B
C
ABC
A
BC
is a triangle of area
4
4
4
with circumcenter
O
O
O
and
M
M
M
is the midpoint of
A
O
AO
A
O
. We choose the points
P
,
Q
P, Q
P
,
Q
on the sides
A
B
AB
A
B
and
A
C
AC
A
C
respectively such that
M
M
M
is at
P
Q
PQ
PQ
and segments
B
C
BC
BC
and
P
Q
PQ
PQ
are parallel. Suppose the area of the triangle
A
P
Q
APQ
A
PQ
is
1
1
1
. Calculate the angle
B
A
C
BAC
B
A
C
.