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National and Regional Contests
Turkey Contests
Turkey MO (2nd round)
1994 Turkey MO (2nd round)
1994 Turkey MO (2nd round)
Part of
Turkey MO (2nd round)
Subcontests
(6)
6
1
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Turkish MO 1994 P6
The incircle of triangle
A
B
C
ABC
A
BC
touches
B
C
BC
BC
at
D
D
D
and
A
C
AC
A
C
at
E
E
E
. Let
K
K
K
be the point on
C
B
CB
CB
with
C
K
=
B
D
CK=BD
C
K
=
B
D
, and
L
L
L
be the point on
C
A
CA
C
A
with
A
E
=
C
L
AE=CL
A
E
=
C
L
. Lines
A
K
AK
A
K
and
B
L
BL
B
L
meet at
P
P
P
. If
Q
Q
Q
is the midpoint of
B
C
BC
BC
,
I
I
I
the incenter, and
G
G
G
the centroid of
△
A
B
C
\triangle ABC
△
A
BC
, show that:
(
a
)
(a)
(
a
)
I
Q
IQ
I
Q
and
A
K
AK
A
K
are parallel,
(
b
)
(b)
(
b
)
the triangles
A
I
G
AIG
A
I
G
and
Q
P
G
QPG
QPG
have equal area.
2
1
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Turkish MO 1994 P2
Let
A
B
C
D
ABCD
A
BC
D
be a cyclic quadrilateral
∠
B
A
D
<
9
0
∘
\angle{BAD}< 90^\circ
∠
B
A
D
<
9
0
∘
and
∠
B
C
A
=
∠
D
C
A
\angle BCA = \angle DCA
∠
BC
A
=
∠
D
C
A
. Point
E
E
E
is taken on segment
D
A
DA
D
A
such that
B
D
=
2
D
E
BD=2DE
B
D
=
2
D
E
. The line through
E
E
E
parallel to
C
D
CD
C
D
intersects the diagonal
A
C
AC
A
C
at
F
F
F
. Prove that
A
C
⋅
B
D
A
B
⋅
F
C
=
2.
\frac{AC\cdot BD}{AB\cdot FC}=2.
A
B
⋅
FC
A
C
⋅
B
D
=
2.
5
1
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Turkish MO 1994 P5
Find the set of all ordered pairs
(
s
,
t
)
(s,t)
(
s
,
t
)
of positive integers such that
t
2
+
1
=
s
(
s
+
1
)
.
t^{2}+1=s(s+1).
t
2
+
1
=
s
(
s
+
1
)
.
3
1
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Turkish MO 1994 P3
Let
n
n
n
blue lines, no two of which are parallel and no three concurrent, be drawn on a plane. An intersection of two blue lines is called a blue point. Through any two blue points that have not already been joined by a blue line, a red line is drawn. An intersection of two red lines is called a red point, and an intersection of red line and a blue line is called a purple point. What is the maximum possible number of purple points?
4
1
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Turkish MO 1994 P4
Let
f
:
R
+
→
R
+
f: \mathbb{R}^{+}\rightarrow \mathbb{R}+
f
:
R
+
→
R
+
be an increasing function. For each
u
∈
R
+
u\in\mathbb{R}^{+}
u
∈
R
+
, we denote
g
(
u
)
=
inf
{
f
(
t
)
+
u
/
t
∣
t
>
0
}
g(u)=\inf\{ f(t)+u/t \mid t>0\}
g
(
u
)
=
in
f
{
f
(
t
)
+
u
/
t
∣
t
>
0
}
. Prove that:
(
a
)
(a)
(
a
)
If
x
≤
g
(
x
y
)
x\leq g(xy)
x
≤
g
(
x
y
)
, then
x
≤
2
f
(
2
y
)
x\leq 2f(2y)
x
≤
2
f
(
2
y
)
;
(
b
)
(b)
(
b
)
If
x
≤
f
(
y
)
x\leq f(y)
x
≤
f
(
y
)
, then
x
≤
2
g
(
x
y
)
x\leq 2g(xy)
x
≤
2
g
(
x
y
)
.
1
1
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Turkish MO 1994 P1
For
n
∈
N
n\in\mathbb{N}
n
∈
N
, let
a
n
a_{n}
a
n
denote the closest integer to
n
\sqrt{n}
n
. Evaluate
∑
n
=
1
∞
1
a
n
3
.
\sum_{n=1}^\infty{\frac{1}{a_{n}^{3}}}.
n
=
1
∑
∞
a
n
3
1
.