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Contests
National and Regional Contests
North Macedonia Contests
Macedonia National Olympiad
2011 Macedonia National Olympiad
2011 Macedonia National Olympiad
Part of
Macedonia National Olympiad
Subcontests
(5)
5
1
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Macedonia National Olympiad 2011 - Problem 5
A table of the type
~
(
n
1
,
n
2
,
.
.
.
,
n
m
)
,
n
1
≥
n
2
≥
.
.
.
≥
n
m
(n_1, n_2, ... , n_m) ,\ n_1 \ge n_2 \ge ... \ge n_m
(
n
1
,
n
2
,
...
,
n
m
)
,
n
1
≥
n
2
≥
...
≥
n
m
~
is defined in the following way:
~
n
1
n_1
n
1
~
squares are ordered horizontally one next to another, then
~
n
2
n_2
n
2
~
squares are ordered horizontally beneath the already ordered
~
n
1
n_1
n
1
~
squares. The procedure continues until a net composed of
~
n
1
n_1
n
1
~
squares in the first row,
~
n
2
n_2
n
2
~
in the second,
~
n
i
n_i
n
i
~
in the
~
i
i
i
-th row is obtained, such that there are totally
~
n
=
n
1
+
n
2
+
.
.
.
+
n
m
n=n_1+n_2+...+n_m
n
=
n
1
+
n
2
+
...
+
n
m
~
squares in the net. The ordered rows form a straight line on the left, as shown in the example. The obtained table is filled with the numbers from
~
1
1
1
~
till
~
n
n
n
~
in a way that the numbers in each row and column become greater from left to right and from top to bottom, respectively. An example of a table of the type
~
(
5
,
4
,
2
,
1
)
(5,4,2,1)
(
5
,
4
,
2
,
1
)
~
and one possible way of filling it is attached to the post. Find the number of ways the table of type
~
(
4
,
3
,
2
)
(4,3,2)
(
4
,
3
,
2
)
~
can be filled.
1
1
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Macedonia National Olympiad 2011 - Problem 1
Let
~
a
,
b
,
c
,
d
>
0
a,\,b,\,c,\,d\, >\, 0
a
,
b
,
c
,
d
>
0
~
and
~
a
+
b
+
c
+
d
=
1
.
a+b+c+d\, =\, 1\, .
a
+
b
+
c
+
d
=
1
.
~
Prove the inequality
1
4
a
+
3
b
+
c
+
1
3
a
+
b
+
4
d
+
1
a
+
4
c
+
3
d
+
1
4
b
+
3
c
+
d
≥
2
.
\frac{1}{4a+3b+c}+\frac{1}{3a+b+4d}+\frac{1}{a+4c+3d}+\frac{1}{4b+3c+d}\; \ge\; 2\, .
4
a
+
3
b
+
c
1
+
3
a
+
b
+
4
d
1
+
a
+
4
c
+
3
d
1
+
4
b
+
3
c
+
d
1
≥
2
.
3
1
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Macedonian National Olympiad 2011 - Problem 3
Find all natural numbers
n
n
n
for which each natural number written with
~
n
−
1
n-1
n
−
1
~
'ones' and one 'seven' is prime.
2
1
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Macedonia National Olympiad 2011 - Problem 2
Acute-angled
~
△
A
B
C
\triangle{ABC}
△
A
BC
~
is given. A line
~
l
l
l
~
parallel to side
~
A
B
AB
A
B
~
passing through vertex
~
C
C
C
~
is drawn. Let the angle bisectors of
~
∠
B
A
C
\angle{BAC}
∠
B
A
C
~
and
~
∠
A
B
C
\angle{ABC}
∠
A
BC
~
intersect the sides
~
B
C
BC
BC
and
~
A
C
AC
A
C
at points
~
D
D
D
~
and
~
F
F
F
, and line
~
l
l
l
~
at points
~
E
E
E
~
and
~
G
G
G
~
respectively. Prove that if
~
D
E
‾
=
G
F
‾
\overline{DE}=\overline{GF}
D
E
=
GF
~
then
~
A
C
‾
=
B
C
‾
.
\overline{AC}=\overline{BC}\, .
A
C
=
BC
.
4
1
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Macedonia National Olympiad 2011 - Problem 4
Find all functions
~
f
:
R
→
R
f: \mathbb{R} \to \mathbb{R}
f
:
R
→
R
~
which satisfy the equation
f
(
x
+
y
f
(
x
)
)
=
f
(
f
(
x
)
)
+
x
f
(
y
)
.
f(x+yf(x))\, =\, f(f(x)) + xf(y)\, .
f
(
x
+
y
f
(
x
))
=
f
(
f
(
x
))
+
x
f
(
y
)
.