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Junior Balkan MO
2012 Junior Balkan MO
2012 Junior Balkan MO
Part of
Junior Balkan MO
Subcontests
(4)
4
1
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JBMO 2012 Problem 4
Find all positive integers
x
,
y
,
z
x,y,z
x
,
y
,
z
and
t
t
t
such that
2
x
3
y
+
5
z
=
7
t
2^x3^y+5^z=7^t
2
x
3
y
+
5
z
=
7
t
.
3
1
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JBMO 2012 Problem 3
On a board there are
n
n
n
nails, each two connected by a rope. Each rope is colored in one of
n
n
n
given distinct colors. For each three distinct colors, there exist three nails connected with ropes of these three colors. a) Can
n
n
n
be
6
6
6
? b) Can
n
n
n
be
7
7
7
?
2
1
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JBMO 2012 Problem 2
Let the circles
k
1
k_1
k
1
and
k
2
k_2
k
2
intersect at two points
A
A
A
and
B
B
B
, and let
t
t
t
be a common tangent of
k
1
k_1
k
1
and
k
2
k_2
k
2
that touches
k
1
k_1
k
1
and
k
2
k_2
k
2
at
M
M
M
and
N
N
N
respectively. If
t
⊥
A
M
t\perp AM
t
⊥
A
M
and
M
N
=
2
A
M
MN=2AM
MN
=
2
A
M
, evaluate the angle
N
M
B
NMB
NMB
.
1
1
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JBMO 2012 Problem 1
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be positive real numbers such that
a
+
b
+
c
=
1
a+b+c=1
a
+
b
+
c
=
1
. Prove that
a
b
+
a
c
+
c
b
+
c
a
+
b
c
+
b
a
+
6
≥
2
2
(
1
−
a
a
+
1
−
b
b
+
1
−
c
c
)
.
\frac {a}{b} + \frac {a}{c} + \frac {c}{b} + \frac {c}{a} + \frac {b}{c} + \frac {b}{a} + 6 \geq 2\sqrt{2}\left (\sqrt{\frac{1-a}{a}} + \sqrt{\frac{1-b}{b}} + \sqrt{\frac{1-c}{c}}\right ).
b
a
+
c
a
+
b
c
+
a
c
+
c
b
+
a
b
+
6
≥
2
2
(
a
1
−
a
+
b
1
−
b
+
c
1
−
c
)
.
When does equality hold?