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Problems
Contests
National and Regional Contests
North Macedonia Contests
Macedonia National Olympiad
2017 Macedonia National Olympiad
2017 Macedonia National Olympiad
Part of
Macedonia National Olympiad
Subcontests
(5)
Problem 5
1
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Macedonia National Olympiad 2017 Problem 5
Let
n
>
1
∈
N
n>1 \in \mathbb{N}
n
>
1
∈
N
and
a
1
,
a
2
,
.
.
.
,
a
n
a_1, a_2, ..., a_n
a
1
,
a
2
,
...
,
a
n
be a sequence of
n
n
n
natural integers. Let:
b
1
=
[
a
2
+
⋯
+
a
n
n
−
1
]
,
b
i
=
[
a
1
+
⋯
+
a
i
−
1
+
a
i
+
1
+
⋯
+
a
n
n
−
1
]
,
b
n
=
[
a
1
+
⋯
+
a
n
−
1
n
−
1
]
b_1 = \left[\frac{a_2 + \cdots + a_n}{n-1}\right], b_i = \left[\frac{a_1 + \cdots + a_{i-1} + a_{i+1} + \cdots + a_n}{n-1}\right], b_n = \left[\frac{a_1 + \cdots + a_{n-1}}{n-1}\right]
b
1
=
[
n
−
1
a
2
+
⋯
+
a
n
]
,
b
i
=
[
n
−
1
a
1
+
⋯
+
a
i
−
1
+
a
i
+
1
+
⋯
+
a
n
]
,
b
n
=
[
n
−
1
a
1
+
⋯
+
a
n
−
1
]
Define a mapping
f
f
f
by
f
(
a
1
,
a
2
,
⋯
a
n
)
=
(
b
1
,
b
2
,
⋯
,
b
n
)
f(a_1,a_2, \cdots a_n) = (b_1,b_2,\cdots,b_n)
f
(
a
1
,
a
2
,
⋯
a
n
)
=
(
b
1
,
b
2
,
⋯
,
b
n
)
.a) Let
g
:
N
→
N
g: \mathbb{N} \to \mathbb{N}
g
:
N
→
N
be a function such that
g
(
1
)
g(1)
g
(
1
)
is the number of different elements in
f
(
a
1
,
a
2
,
⋯
a
n
)
f(a_1,a_2, \cdots a_n)
f
(
a
1
,
a
2
,
⋯
a
n
)
and
g
(
m
)
g(m)
g
(
m
)
is the number od different elements in
f
m
(
a
1
,
a
2
,
⋯
a
n
)
=
f
(
f
m
−
1
(
a
1
,
a
2
,
⋯
a
n
)
)
;
m
>
1
f^m(a_1,a_2, \cdots a_n) = f(f^{m-1}(a_1,a_2, \cdots a_n)); m>1
f
m
(
a
1
,
a
2
,
⋯
a
n
)
=
f
(
f
m
−
1
(
a
1
,
a
2
,
⋯
a
n
))
;
m
>
1
. Prove that
∃
k
0
∈
N
\exists k_0 \in \mathbb{N}
∃
k
0
∈
N
s.t. for
m
≥
k
0
m \ge k_0
m
≥
k
0
the function
g
(
m
)
g(m)
g
(
m
)
is periodic.b) Prove that
∑
m
=
1
k
g
(
m
)
m
(
m
+
1
)
<
C
\sum_{m=1}^k \frac{g(m)}{m(m+1)} < C
∑
m
=
1
k
m
(
m
+
1
)
g
(
m
)
<
C
for all
k
∈
N
k \in \mathbb{N}
k
∈
N
, where
C
C
C
is a function that doesn't depend on
k
k
k
.
Problem 4
1
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Macedonia National Olympiad 2017 Problem 4
Let
O
O
O
be the circumcenter of the acute triangle
A
B
C
ABC
A
BC
(
A
B
<
A
C
AB < AC
A
B
<
A
C
). Let
A
1
A_1
A
1
and
P
P
P
be the feet of the perpendicular lines drawn from
A
A
A
and
O
O
O
to
B
C
BC
BC
, respectively. The lines
B
O
BO
BO
and
C
O
CO
CO
intersect
A
A
1
AA_1
A
A
1
in
D
D
D
and
E
E
E
, respectively. Let
F
F
F
be the second intersection point of
⊙
A
B
D
\odot ABD
⊙
A
B
D
and
⊙
A
C
E
\odot ACE
⊙
A
CE
. Prove that the angle bisector od
∠
F
A
P
\angle FAP
∠
F
A
P
passes through the incenter of
△
A
B
C
\triangle ABC
△
A
BC
.
Problem 3
1
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Macedonia National Olympiad 2017 Problem 3 Inequality
Let
x
,
y
,
z
∈
R
x,y,z \in \mathbb{R}
x
,
y
,
z
∈
R
such that
x
y
z
=
1
xyz = 1
x
yz
=
1
. Prove that:
(
x
4
+
z
2
y
2
)
(
y
4
+
x
2
z
2
)
(
z
4
+
y
2
x
2
)
≥
(
x
2
y
+
1
)
(
y
2
z
+
1
)
(
z
2
x
+
1
)
.
\left(x^4 + \frac{z^2}{y^2}\right)\left(y^4 + \frac{x^2}{z^2}\right)\left(z^4 + \frac{y^2}{x^2}\right) \ge \left(\frac{x^2}{y} + 1 \right)\left(\frac{y^2}{z} + 1 \right)\left(\frac{z^2}{x} + 1 \right).
(
x
4
+
y
2
z
2
)
(
y
4
+
z
2
x
2
)
(
z
4
+
x
2
y
2
)
≥
(
y
x
2
+
1
)
(
z
y
2
+
1
)
(
x
z
2
+
1
)
.
Problem 2
1
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Macedonia National Olympiad 2017 Problem 2
Find all natural integers
n
n
n
such that
(
n
3
+
39
n
−
2
)
n
!
+
17
⋅
2
1
n
+
5
(n^3 + 39n - 2)n! + 17\cdot 21^n + 5
(
n
3
+
39
n
−
2
)
n
!
+
17
⋅
2
1
n
+
5
is a square.
Problem 1
1
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Macedonia National Olympiad 2017 Problem 1
Find all functions
f
:
N
→
N
f:\mathbb{N} \to \mathbb{N}
f
:
N
→
N
such that for each natural integer
n
>
1
n>1
n
>
1
and for all
x
,
y
∈
N
x,y \in \mathbb{N}
x
,
y
∈
N
the following holds:
f
(
x
+
y
)
=
f
(
x
)
+
f
(
y
)
+
∑
k
=
1
n
−
1
(
n
k
)
x
n
−
k
y
k
f(x+y) = f(x) + f(y) + \sum_{k=1}^{n-1} \binom{n}{k}x^{n-k}y^k
f
(
x
+
y
)
=
f
(
x
)
+
f
(
y
)
+
k
=
1
∑
n
−
1
(
k
n
)
x
n
−
k
y
k