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International Contests
Junior Balkan MO
2016 Junior Balkan MO
2016 Junior Balkan MO
Part of
Junior Balkan MO
Subcontests
(4)
4
1
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JBMO 2016 Problem 4
A
5
×
5
5 \times 5
5
×
5
table is called regular f each of its cells contains one of four pairwise distinct real numbers,such that each of them occurs exactly one in every
2
×
2
2 \times 2
2
×
2
subtable.The sum of all numbers of a regular table is called the total sum of the table.With any four numbers,one constructs all possible regular tables,computes their total sums and counts the distinct outcomes.Determine the maximum possible count.
3
1
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JBMO 2016 Problem 3
Find all triplets of integers
(
a
,
b
,
c
)
(a,b,c)
(
a
,
b
,
c
)
such that the number
N
=
(
a
−
b
)
(
b
−
c
)
(
c
−
a
)
2
+
2
N = \frac{(a-b)(b-c)(c-a)}{2} + 2
N
=
2
(
a
−
b
)
(
b
−
c
)
(
c
−
a
)
+
2
is a power of
2016
2016
2016
.(A power of
2016
2016
2016
is an integer of form
201
6
n
2016^n
201
6
n
,where n is a non-negative integer.)
2
1
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JBMO 2016 Problem 2
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be positive real numbers.Prove that
8
(
a
+
b
)
2
+
4
a
b
c
+
8
(
b
+
c
)
2
+
4
a
b
c
+
8
(
a
+
c
)
2
+
4
a
b
c
+
a
2
+
b
2
+
c
2
≥
8
a
+
3
+
8
b
+
3
+
8
c
+
3
\frac{8}{(a+b)^2 + 4abc} + \frac{8}{(b+c)^2 + 4abc} + \frac{8}{(a+c)^2 + 4abc} + a^2 + b^2 + c ^2 \ge \frac{8}{a+3} + \frac{8}{b+3} + \frac{8}{c+3}
(
a
+
b
)
2
+
4
ab
c
8
+
(
b
+
c
)
2
+
4
ab
c
8
+
(
a
+
c
)
2
+
4
ab
c
8
+
a
2
+
b
2
+
c
2
≥
a
+
3
8
+
b
+
3
8
+
c
+
3
8
.
1
1
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JBMO 2016 Problem 1
A trapezoid
A
B
C
D
ABCD
A
BC
D
(
A
B
∣
∣
C
F
AB || CF
A
B
∣∣
CF
,
A
B
>
C
D
AB > CD
A
B
>
C
D
) is circumscribed.The incircle of the triangle
A
B
C
ABC
A
BC
touches the lines
A
B
AB
A
B
and
A
C
AC
A
C
at the points
M
M
M
and
N
N
N
,respectively.Prove that the incenter of the trapezoid
A
B
C
D
ABCD
A
BC
D
lies on the line
M
N
MN
MN
.