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Contests
National and Regional Contests
North Macedonia Contests
JBMO TST - Macedonia
2016 Macedonia JBMO TST
2016 Macedonia JBMO TST
Part of
JBMO TST - Macedonia
Subcontests
(5)
5
1
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2016 JBMO TST- Macedonia, problem 5
Solve the following equation in the set of positive integers
x
+
y
2
+
(
G
C
D
(
x
,
y
)
)
2
=
x
y
⋅
G
C
D
(
x
,
y
)
x + y^2 + (GCD(x, y))^2 = xy \cdot GCD(x, y)
x
+
y
2
+
(
GC
D
(
x
,
y
)
)
2
=
x
y
⋅
GC
D
(
x
,
y
)
.
4
1
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2016 JBMO TST- Macedonia, problem 4
Let
x
x
x
,
y
y
y
, and
z
z
z
be positive real numbers. Prove that
x
y
x
2
+
y
2
+
2
z
2
+
y
z
y
2
+
z
2
+
2
x
2
+
z
x
z
2
+
x
2
+
2
y
2
≤
3
2
\sqrt {\frac {xy}{x^2 + y^2 + 2z^2}} + \sqrt {\frac {yz}{y^2 + z^2 + 2x^2}}+\sqrt {\frac {zx}{z^2 + x^2 + 2y^2}} \le \frac{3}{2}
x
2
+
y
2
+
2
z
2
x
y
+
y
2
+
z
2
+
2
x
2
yz
+
z
2
+
x
2
+
2
y
2
z
x
≤
2
3
.When does equality hold?
3
1
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2016 JBMO TST- Macedonia, problem 3
We are given a
4
×
4
4\times4
4
×
4
square, consisting of
16
16
16
squares with side length of
1
1
1
. Every
1
×
1
1\times1
1
×
1
square inside the square has a non-negative integer entry such that the sum of any five squares that can be covered with the figures down below (the figures can be moved and rotated) equals
5
5
5
. What is the greatest number of different numbers that can be used to cover the square?
2
1
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2016 JBMO TST- Macedonia, problem 2
Let
A
B
C
D
ABCD
A
BC
D
be a parallelogram and let
E
E
E
,
F
F
F
,
G
G
G
, and
H
H
H
be the midpoints of sides
A
B
AB
A
B
,
B
C
BC
BC
,
C
D
CD
C
D
, and
D
A
DA
D
A
, respectively. If
B
H
∩
A
C
=
I
BH \cap AC = I
B
H
∩
A
C
=
I
,
B
D
∩
E
C
=
J
BD \cap EC = J
B
D
∩
EC
=
J
,
A
C
∩
D
F
=
K
AC \cap DF = K
A
C
∩
D
F
=
K
, and
A
G
∩
B
D
=
L
AG \cap BD = L
A
G
∩
B
D
=
L
, prove that the quadrilateral
I
J
K
L
IJKL
I
J
K
L
is a parallelogram.
1
1
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2016 JBMO TST- Macedonia, problem 1
Solve the following equation in the set of integers
x
1
4
+
x
2
4
+
.
.
.
+
x
14
4
=
201
6
3
−
1
x_{1}^4 + x_{2}^4 +...+ x_{14}^4=2016^3 - 1
x
1
4
+
x
2
4
+
...
+
x
14
4
=
201
6
3
−
1
.