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Problems
Contests
National and Regional Contests
Chile Contests
Chile National Olympiad
2010 Chile National Olympiad
2010 Chile National Olympiad
Part of
Chile National Olympiad
Subcontests
(6)
6
1
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finite no of equal circles inside equilateral
Prove that in the interior of an equilateral triangle with side
a
a
a
you can put a finite number of equal circles that do not overlap, with radius
r
=
a
2010
r = \frac{a}{2010}
r
=
2010
a
, so that the sum of their areas is greater than
17
3
80
\frac{17\sqrt3}{80}
80
17
3
a
2
^2
2
.
4
1
Hide problems
m + n\sqrt2 = (1 +\sqrt2 )^{2010}
Let
m
,
n
m, n
m
,
n
integers such that satisfy
m
+
n
2
=
(
1
+
2
)
2010
.
m + n\sqrt2 = \left(1 +\sqrt2\right)^{2010} .
m
+
n
2
=
(
1
+
2
)
2010
.
Find the remainder that is obtained when dividing
n
n
n
by
5
5
5
.
2
1
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compare 10^{10^{10^{10}}} with (10^{10})!
Determine which of the following numbers is greater
1
0
1
0
1
0
10
,
(
1
0
10
)
!
10^{10^{10^{10}}}, (10^{10})!
1
0
1
0
1
0
10
,
(
1
0
10
)!
1
1
Hide problems
a- b, 2a + 2b + 1, 3a + 3b + 1 perfect sqaures if 2a^2 + a = 3b^2 + b
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.
5
1
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d(A,ll)> d(P, AQ), distances inequality with 3 collinear points Chile 2010 L2 P5
Consider a line
ℓ
\ell
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 different points in
ℓ
\ell
ℓ
. Let
A
A
A
be a point that is not in
ℓ
\ell
ℓ
. Show that there is
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
so that the distance from
A
A
A
to
ℓ
\ell
ℓ
is greater than the distance from
P
P
P
to the line that passes through
A
A
A
and
Q
Q
Q
.
3
1
Hide problems
incenter lies on a segment connecting midpoints (Chile 2010 L2 P3)
The sides
B
C
,
C
A
BC, CA
BC
,
C
A
, and
A
B
AB
A
B
of a triangle
A
B
C
ABC
A
BC
are tangent to a circle at points
X
,
Y
,
Z
X, Y, Z
X
,
Y
,
Z
respectively. Show that the center of such a circle is on the line that passes through the midpoints of
B
C
BC
BC
and
A
X
AX
A
X
.