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Chile Junior Math Olympiad
2006 Chile Junior Math Olympiad
2006 Chile Junior Math Olympiad
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Chile Junior Math Olympiad
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2006 Chile NMO Juniors XVIII
[url=https://artofproblemsolving.com/community/c1068820h2917798p26064010]p1. Consider three circles of radius one tangent to each other. Find the radius of each of the circles tangent to the three given circles. p2. A three-digit number is called balanced if any of its numbers is the average of the others. For example
528
528
528
is balanced since
5
=
2
+
8
2
5 =\frac{2+8}{2}
5
=
2
2
+
8
. How many balanced three-digit numbers are there? [url=https://artofproblemsolving.com/community/c1068820h2917801p26064051]p3.
A
B
=
3
AB = 3
A
B
=
3
,
A
C
=
8
AC = 8
A
C
=
8
,
A
1
A_1
A
1
midpoint of
A
C
AC
A
C
,
B
A
1
∥
B
1
A
2
∥
B
2
A
3
BA_1 \parallel B_1A_2 \parallel B_2A_3
B
A
1
∥
B
1
A
2
∥
B
2
A
3
, ...
A
B
⊥
A
C
AB\perp AC
A
B
⊥
A
C
,
A
1
B
1
⊥
A
C
A_1B_1\perp AC
A
1
B
1
⊥
A
C
,
A
2
B
2
⊥
A
C
A_2B_2\perp AC
A
2
B
2
⊥
A
C
,
A
3
B
3
⊥
A
C
A_3B_3\perp AC
A
3
B
3
⊥
A
C
... https://cdn.artofproblemsolving.com/attachments/8/3/5fdbdaec663ce80a031c4ebbaa1418085d6de7.jpg Find
2
9
(
B
A
1
+
B
1
A
2
+
B
2
A
3
+
.
.
.
+
B
8
A
9
)
2^9 (BA_1 + B_1A_2 + B_2A_3 + ... + B_8A_9)
2
9
(
B
A
1
+
B
1
A
2
+
B
2
A
3
+
...
+
B
8
A
9
)
[url=https://artofproblemsolving.com/community/c1068820h2917799p26064022]p4. Let
A
B
C
ABC
A
BC
be a triangle right at
C
C
C
,
a
a
a
and
b
b
b
the lengths of its legs and
r
r
r
the radius of the circle that is tangent to the legs and that has the center
O
O
O
at the hypotenuse. Prove that
1
r
=
1
a
+
1
b
\frac{1}{r}=\frac{1}{a}+\frac{1}{b}
r
1
=
a
1
+
b
1
. [url=https://artofproblemsolving.com/community/c1068820h2917800p26064029]p5.
A
B
C
D
ABCD
A
BC
D
is a square,
E
E
E
is the midpoint of
B
C
BC
BC
,
F
∈
D
E
F \in DE
F
∈
D
E
such that
A
F
⊥
D
E
AF \perp DE
A
F
⊥
D
E
. Prove that
∠
C
D
E
=
∠
B
F
E
\angle CDE = \angle BFE
∠
C
D
E
=
∠
BFE
. p6. Let
x
x
x
be a real number,
[
x
]
[x]
[
x
]
its integer part,
{
x
}
=
x
−
[
x
]
\{x\} = x-[x]
{
x
}
=
x
−
[
x
]
. Find the largest number
C
C
C
such that for all x the inequality
x
2
≥
C
⋅
[
x
]
⋅
{
x
}
x^2\ge C \cdot [x] \cdot \{x\}
x
2
≥
C
⋅
[
x
]
⋅
{
x
}
holds.