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Chile Junior Math Olympiad
2010 Chile Junior Math Olympiad
2010 Chile Junior Math Olympiad
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Chile Junior Math Olympiad
Subcontests
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2010 Chile NMO Juniors XXII
p1. Determine which of the following numbers is greater
1
0
1
0
1
0
10
,
(
1
0
10
)
(
1
0
10
)
.
10^{10^{10^{10}}}, (10^{10})^{(10^{10})}.
1
0
1
0
1
0
10
,
(
1
0
10
)
(
1
0
10
)
.
[url=https://artofproblemsolving.com/community/c1068820h2935461p26269201]p2. The integers
a
,
b
a, b
a
,
b
satisfy the following identity
2
a
2
+
a
=
3
b
2
+
b
.
2a^2 + a = 3b^2 + b.
2
a
2
+
a
=
3
b
2
+
b
.
Prove that
a
−
b
a- b
a
−
b
,
2
a
+
2
b
+
1
2a + 2b + 1
2
a
+
2
b
+
1
, and
3
a
+
3
b
+
1
3a + 3b + 1
3
a
+
3
b
+
1
are perfect squares. [url=https://artofproblemsolving.com/community/c4h1845719p12426408]p3. Let
A
B
C
D
ABCD
A
BC
D
be a square and
M
M
M
be its center. Consider the point
E
E
E
on line
A
C
AC
A
C
such that
∣
M
C
∣
=
∣
C
E
∣
| MC | = | CE |
∣
MC
∣
=
∣
CE
∣
. Let
S
S
S
be the circle circumscribed to triangle
△
E
D
B
\triangle EDB
△
E
D
B
. Show that
S
S
S
passes through the midpoint of
A
M
AM
A
M
. p4. Find the sum of all positive divisors of the number
2.010.000.000
2.010.000.000
2.010.000.000
. p5. It is known that the sides and the diagonal of a rectangle are integers. Prove that the area of the rectangle is divisible by
12
12
12
. [url=https://artofproblemsolving.com/community/c4h1845722p12426437]p6. Consider a line
L
L
L
in the plane and let
B
1
,
B
2
,
B
3
B_1, B_2, B_3
B
1
,
B
2
,
B
3
be points different in
L
L
L
. Let
A
A
A
be a point that does not lie in
L
L
L
. Show that there are
P
,
Q
P, Q
P
,
Q
in
{
B
1
,
B
2
,
B
3
}
\{B_1, B_2, B_3\}
{
B
1
,
B
2
,
B
3
}
with
P
≠
Q
P \ne Q
P
=
Q
such that the distance from
A
A
A
to
L
L
L
to be greater than the distance from
P
P
P
to the line that passes through
A
A
A
and
Q
Q
Q
.PS. Problem 2 was also proposed as Seniors P1.