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Contests
National and Regional Contests
Turkey Contests
Turkey MO (2nd round)
1995 Turkey MO (2nd round)
1995 Turkey MO (2nd round)
Part of
Turkey MO (2nd round)
Subcontests
(6)
5
1
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Turkish MO 1995 P5
Let
t
(
A
)
t(A)
t
(
A
)
denote the sum of elements of a nonempty set
A
A
A
of integers, and define
t
(
∅
)
=
0
t(\emptyset)=0
t
(
∅
)
=
0
. Find a set
X
X
X
of positive integers such that for every integers
k
k
k
there is a unique ordered pair of disjoint subsets
(
A
k
,
B
k
)
(A_{k},B_{k})
(
A
k
,
B
k
)
of
X
X
X
such that
t
(
A
k
)
−
t
(
B
k
)
=
k
t(A_{k})-t(B_{k}) = k
t
(
A
k
)
−
t
(
B
k
)
=
k
.
4
1
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Turkish MO 1995 P4
In a triangle
A
B
C
ABC
A
BC
with
A
B
≠
A
C
AB\neq AC
A
B
=
A
C
, the internal and external bisectors of angle
A
A
A
meet the line
B
C
BC
BC
at
D
D
D
and
E
E
E
respectively. If the feet of the perpendiculars from a point
F
F
F
on the circle with diameter
D
E
DE
D
E
to
B
C
,
C
A
,
A
B
BC,CA,AB
BC
,
C
A
,
A
B
are
K
,
L
,
M
K,L,M
K
,
L
,
M
, respectively, show that
K
L
=
K
M
KL=KM
K
L
=
K
M
.
6
1
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Turkish MO 1995 P6
Find all surjective functions
f
:
N
→
N
f: \mathbb{N}\rightarrow \mathbb{N}
f
:
N
→
N
such that for all
m
,
n
∈
N
m,n\in \mathbb{N}
m
,
n
∈
N
f(m)\mid f(n) \mbox{ if and only if }m\mid n.
3
1
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Turkish MO 1995 P3
Let
A
A
A
be a real number and
(
a
n
)
(a_{n})
(
a
n
)
be a sequence of real numbers such that
a
1
=
1
a_{1}=1
a
1
=
1
and 1<\frac{a_{n+1}}{a_{n}}\leq A \mbox{ for all }n\in\mathbb{N}.
(
a
)
(a)
(
a
)
Show that there is a unique non-decreasing surjective function
f
:
N
→
N
f: \mathbb{N}\rightarrow \mathbb{N}
f
:
N
→
N
such that
1
<
A
k
(
n
)
/
a
n
≤
A
1<A^{k(n)}/a_{n}\leq A
1
<
A
k
(
n
)
/
a
n
≤
A
for all
n
∈
N
n\in \mathbb{N}
n
∈
N
.
(
b
)
(b)
(
b
)
If
k
k
k
takes every value at most
m
m
m
times, show that there is a real number
C
>
1
C>1
C
>
1
such that
A
a
n
≥
C
n
Aa_{n}\geq C^{n}
A
a
n
≥
C
n
for all
n
∈
N
n\in \mathbb{N}
n
∈
N
.
2
1
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Turkish MO 1995 P2
Let
A
B
C
ABC
A
BC
be an acute triangle and let
k
1
,
k
2
,
k
3
k_{1},k_{2},k_{3}
k
1
,
k
2
,
k
3
be the circles with diameters
B
C
,
C
A
,
A
B
BC,CA,AB
BC
,
C
A
,
A
B
, respectively. Let
K
K
K
be the radical center of these circles. Segments
A
K
,
C
K
,
B
K
AK,CK,BK
A
K
,
C
K
,
B
K
meet
k
1
,
k
2
,
k
3
k_{1},k_{2},k_{3}
k
1
,
k
2
,
k
3
again at
D
,
E
,
F
D,E,F
D
,
E
,
F
, respectively. If the areas of triangles
A
B
C
,
D
B
C
,
E
C
A
,
F
A
B
ABC,DBC,ECA,FAB
A
BC
,
D
BC
,
EC
A
,
F
A
B
are
u
,
x
,
y
,
z
u,x,y,z
u
,
x
,
y
,
z
, respectively, prove that
u
2
=
x
2
+
y
2
+
z
2
.
u^{2}=x^{2}+y^{2}+z^{2}.
u
2
=
x
2
+
y
2
+
z
2
.
1
1
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Turkish MO 1995 P1
Let
m
1
,
m
2
,
…
,
m
k
m_{1},m_{2},\ldots,m_{k}
m
1
,
m
2
,
…
,
m
k
be integers with
2
≤
m
1
2\leq m_{1}
2
≤
m
1
and
2
m
1
≤
m
i
+
1
2m_{1}\leq m_{i+1}
2
m
1
≤
m
i
+
1
for all
i
i
i
. Show that for any integers
a
1
,
a
2
,
…
,
a
k
a_{1},a_{2},\ldots,a_{k}
a
1
,
a
2
,
…
,
a
k
there are infinitely many integers
x
x
x
which do not satisfy any of the congruences
x
≡
a
i
(
m
o
d
m
i
)
,
i
=
1
,
2
,
…
k
.
x\equiv a_{i}\ (\bmod \ m_{i}),\ i=1,2,\ldots k.
x
≡
a
i
(
mod
m
i
)
,
i
=
1
,
2
,
…
k
.