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Contests
National and Regional Contests
North Macedonia Contests
Macedonia National Olympiad
2016 Macedonia National Olympiad
2016 Macedonia National Olympiad
Part of
Macedonia National Olympiad
Subcontests
(5)
Problem 5
1
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Macedonia National Olympiad 2016 Problem 5
Let
n
≥
3
n\ge3
n
≥
3
and
a
1
,
a
2
,
.
.
.
,
a
n
∈
R
+
a_1,a_2,...,a_n \in \mathbb{R^{+}}
a
1
,
a
2
,
...
,
a
n
∈
R
+
, such that
1
1
+
a
1
4
+
1
1
+
a
2
4
+
.
.
.
+
1
1
+
a
n
4
=
1
\frac{1}{1+a_1^4} + \frac{1}{1+a_2^4} + ... + \frac{1}{1+a_n^4} = 1
1
+
a
1
4
1
+
1
+
a
2
4
1
+
...
+
1
+
a
n
4
1
=
1
. Prove that:
a
1
a
2
.
.
.
a
n
≥
(
n
−
1
)
n
4
a_1a_2...a_n \ge (n-1)^{\frac n4}
a
1
a
2
...
a
n
≥
(
n
−
1
)
4
n
Problem 4
1
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Macedonia National Olympiad 2016 Problem 4
A segment
A
B
AB
A
B
is given and it's midpoint
K
K
K
. On the perpendicular line to
A
B
AB
A
B
, passing through
K
K
K
a point
C
C
C
, different from
K
K
K
is chosen. Let
N
N
N
be the intersection of
A
C
AC
A
C
and the line passing through
B
B
B
and the midpoint of
C
K
CK
C
K
. Let
U
U
U
be the intersection point of
A
B
AB
A
B
and the line passing through
C
C
C
and
L
L
L
, the midpoint of
B
N
BN
BN
. Prove that the ratio of the areas of the triangles
C
N
L
CNL
CN
L
and
B
U
L
BUL
B
UL
, is independent of the choice of the point
C
C
C
.
Problem 3
1
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Macedonia National Olympiad 2016 Problem 3
Solve the equation in the set of natural numbers
x
y
z
+
y
z
t
+
x
z
t
+
x
y
t
=
x
y
z
t
+
3
xyz+yzt+xzt+xyt = xyzt + 3
x
yz
+
yz
t
+
x
z
t
+
x
y
t
=
x
yz
t
+
3
Problem 2
1
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Macedonia National Olympiad 2016 Problem 2
A magic square is a square with side 3 consisting of 9 unit squares, such that the numbers written in the unit squares (one number in each square) satisfy the following property: the sum of the numbers in each row is equal to the sum of the numbers in each column and is equal to the sum of all the numbers written in any of the two diagonals.A rectangle with sides
m
≥
3
m\ge3
m
≥
3
and
n
≥
3
n\ge3
n
≥
3
consists of
m
n
mn
mn
unit squares. If in each of those unit squares exactly one number is written, such that any square with side
3
3
3
is a magic square, then find the number of most different numbers that can be written in that rectangle.
Problem 1
1
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Macedonia National Olympiad 2016 Problem 1
Solve the equation in the set of natural numbers
1
+
x
z
+
y
z
=
L
C
M
(
x
z
,
y
z
)
1+x^z + y^z = LCM(x^z,y^z)
1
+
x
z
+
y
z
=
L
CM
(
x
z
,
y
z
)