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Problems
Contests
National and Regional Contests
North Macedonia Contests
Macedonia National Olympiad
2013 Macedonia National Olympiad
2013 Macedonia National Olympiad
Part of
Macedonia National Olympiad
Subcontests
(5)
5
1
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Problem 5
An arbitrary triangle ABC is given. There are 2 lines, p and q, that are not parallel to each other and they are not perpendicular to the sides of the triangle. The perpendicular lines through points A, B and C to line p we denote with
p
a
,
p
b
,
p
c
p_a, p_b, p_c
p
a
,
p
b
,
p
c
and the perpendicular lines to line q we denote with
q
a
,
q
b
,
q
c
q_a, q_b, q_c
q
a
,
q
b
,
q
c
. Let the intersection points of the lines
p
a
,
q
a
,
p
b
,
q
b
,
p
c
p_a, q_a, p_b, q_b, p_c
p
a
,
q
a
,
p
b
,
q
b
,
p
c
and
q
c
q_c
q
c
with
q
b
,
p
b
,
q
c
,
p
c
,
q
a
q_b, p_b, q_c, p_c, q_a
q
b
,
p
b
,
q
c
,
p
c
,
q
a
and
p
a
p_a
p
a
are
K
,
L
,
P
,
Q
,
N
K, L, P, Q, N
K
,
L
,
P
,
Q
,
N
and
M
M
M
. Prove that
K
L
,
M
N
KL, MN
K
L
,
MN
and
P
Q
PQ
PQ
intersect in one point.
4
1
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Problem 4
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be positive real numbers such that
a
4
+
b
4
+
c
4
=
3
a^4+b^4+c^4=3
a
4
+
b
4
+
c
4
=
3
. Prove that
9
a
2
+
b
4
+
c
6
+
9
a
4
+
b
6
+
c
2
+
9
a
6
+
b
2
+
c
4
≤
a
6
+
b
6
+
c
6
+
6
\frac{9}{a^2+b^4+c^6}+\frac{9}{a^4+b^6+c^2}+\frac{9}{a^6+b^2+c^4}\leq\ a^6+b^6+c^6+6
a
2
+
b
4
+
c
6
9
+
a
4
+
b
6
+
c
2
9
+
a
6
+
b
2
+
c
4
9
≤
a
6
+
b
6
+
c
6
+
6
3
1
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Problem 3
Acute angle triangle is given such that
B
C
BC
BC
is the longest side. Let
E
E
E
and
G
G
G
be the intersection points from the altitude from
A
A
A
to
B
C
BC
BC
with the circumscribed circle of triangle
A
B
C
ABC
A
BC
and
B
C
BC
BC
respectively. Let the center
O
O
O
of this circle is positioned on the perpendicular line from
A
A
A
to
B
E
BE
BE
. Let
E
M
EM
EM
be perpendicular to
A
C
AC
A
C
and
E
F
EF
EF
be perpendicular to
A
B
AB
A
B
. Prove that the area of
F
B
E
G
FBEG
FBEG
is greater than the area of
M
F
E
MFE
MFE
.
2
1
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Problem 2
2
n
2^n
2
n
coins are given to a couple of kids. Interchange of the coins occurs when some of the kids has at least half of all the coins. Then from the coins of one of those kids to the all other kids are given that much coins as the kid already had. In case when all the coins are at one kid there is no possibility for interchange. What is the greatest possible number of consecutive interchanges? (
n
n
n
is natural number)
1
1
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Problem 1
Let
p
,
q
,
r
p,q,r
p
,
q
,
r
be prime numbers. Solve the equation
p
2
q
+
q
2
p
=
r
p^{2q}+q^{2p}=r
p
2
q
+
q
2
p
=
r