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Contests
National and Regional Contests
Turkey Contests
Turkey MO (2nd round)
2004 Turkey MO (2nd round)
2004 Turkey MO (2nd round)
Part of
Turkey MO (2nd round)
Subcontests
(6)
6
1
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Turkey NMO 2004, Find the number of elements of K(2004,2004)
Define
K
(
n
,
0
)
=
∅
K(n,0)=\varnothing
K
(
n
,
0
)
=
∅
and, for all nonnegative integers m and n,
K
(
n
,
m
+
1
)
=
{
k
∣
1
≤
k
≤
n
and
K
(
k
,
m
)
∩
K
(
n
−
k
,
m
)
=
∅
}
K(n,m+1)=\left\{ \left. k \right|\text{ }1\le k\le n\text{ and }K(k,m)\cap K(n-k,m)=\varnothing \right\}
K
(
n
,
m
+
1
)
=
{
k
∣
1
≤
k
≤
n
and
K
(
k
,
m
)
∩
K
(
n
−
k
,
m
)
=
∅
}
. Find the number of elements of
K
(
2004
,
2004
)
K(2004,2004)
K
(
2004
,
2004
)
.
4
1
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Turkey NMO 2004 P.4, A Function Problem
Find all functions
f
:
Z
→
Z
f:\mathbb{Z}\to \mathbb{Z}
f
:
Z
→
Z
satisfying the condition
f
(
n
)
−
f
(
n
+
f
(
m
)
)
=
m
f(n)-f(n+f(m))=m
f
(
n
)
−
f
(
n
+
f
(
m
))
=
m
for all
m
,
n
∈
Z
m,n\in \mathbb{Z}
m
,
n
∈
Z
5
1
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Turkey NMO 2004 P.5, Nice Geometric Inequality
The excircle of a triangle
A
B
C
ABC
A
BC
corresponding to
A
A
A
touches the lines
B
C
,
C
A
,
A
B
BC,CA,AB
BC
,
C
A
,
A
B
at
A
1
,
B
1
,
C
1
A_1,B_1,C_1
A
1
,
B
1
,
C
1
, respectively. The excircle corresponding to
B
B
B
touches
B
C
,
C
A
,
A
B
BC,CA,AB
BC
,
C
A
,
A
B
at
A
2
,
B
2
,
C
2
A_2,B_2,C_2
A
2
,
B
2
,
C
2
, and the excircle corresponding to
C
C
C
touches
B
C
,
C
A
,
A
B
BC,CA,AB
BC
,
C
A
,
A
B
at
A
3
,
B
3
,
C
3
A_3,B_3,C_3
A
3
,
B
3
,
C
3
, respectively. Find the maximum possible value of the ratio of the sum of the perimeters of
△
A
1
B
1
C
1
\triangle A_1B_1C_1
△
A
1
B
1
C
1
,
△
A
2
B
2
C
2
\triangle A_2B_2C_2
△
A
2
B
2
C
2
and
△
A
3
B
3
C
3
\triangle A_3B_3C_3
△
A
3
B
3
C
3
to the circumradius of
△
A
B
C
\triangle ABC
△
A
BC
.
1
1
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Turkey NMO 2004 P.1, Easy Problem
In a triangle
△
A
B
C
\triangle ABC
△
A
BC
with
∠
B
>
∠
C
\angle B>\angle C
∠
B
>
∠
C
, the altitude, the angle bisector, and the median from
A
A
A
intersect
B
C
BC
BC
at
H
,
L
H, L
H
,
L
and
D
D
D
, respectively. Show that
∠
H
A
L
=
∠
D
A
L
\angle HAL=\angle DAL
∠
H
A
L
=
∠
D
A
L
if and only if
∠
B
A
C
=
9
0
∘
\angle BAC=90^{\circ}
∠
B
A
C
=
9
0
∘
.
2
1
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Flights between cities (Turkey 2004)
Two-way flights are operated between
80
80
80
cities in such a way that each city is connected to at least
7
7
7
other cities by a direct flight and any two cities are connected by a finite sequence of flights. Find the smallest
k
k
k
such that for any such arrangement of flights it is possible to travel from any city to any other city by a sequence of at most
k
k
k
flights.
3
1
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Turkey NMO-2004 P.3, nice problem about diophantine equation
(a) Determine if exist an integer
n
n
n
such that
n
2
−
k
n^2 -k
n
2
−
k
has exactly
10
10
10
positive divisors for each
k
=
1
,
2
,
3.
k = 1, 2, 3.
k
=
1
,
2
,
3.
(b) Show that the number of positive divisors of
n
2
−
4
n^2 -4
n
2
−
4
is not
10
10
10
for any integer
n
.
n.
n
.