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Contests
National and Regional Contests
Turkey Contests
Turkey Team Selection Test
2013 Turkey Team Selection Test
2013 Turkey Team Selection Test
Part of
Turkey Team Selection Test
Subcontests
(3)
1
1
Hide problems
Concyclic Points
Let
E
E
E
be intersection of the diagonals of convex quadrilateral
A
B
C
D
ABCD
A
BC
D
. It is given that
m
(
E
D
C
^
)
=
m
(
D
E
C
^
)
=
m
(
B
A
D
^
)
m(\widehat{EDC}) = m(\widehat{DEC})=m(\widehat{BAD})
m
(
E
D
C
)
=
m
(
D
EC
)
=
m
(
B
A
D
)
. If
F
F
F
is a point on
[
B
C
]
[BC]
[
BC
]
such that
m
(
B
A
F
^
)
+
m
(
E
B
F
^
)
=
m
(
B
F
E
^
)
m(\widehat{BAF}) + m(\widehat{EBF})=m(\widehat{BFE})
m
(
B
A
F
)
+
m
(
EBF
)
=
m
(
BFE
)
, show that
A
A
A
,
B
B
B
,
F
F
F
,
D
D
D
are concyclic.
2
3
Hide problems
KM.LN=BM.CN
Let the incircle of the triangle
A
B
C
ABC
A
BC
touch
[
B
C
]
[BC]
[
BC
]
at
D
D
D
and
I
I
I
be the incenter of the triangle. Let
T
T
T
be midpoint of
[
I
D
]
[ID]
[
I
D
]
. Let the perpendicular from
I
I
I
to
A
D
AD
A
D
meet
A
B
AB
A
B
and
A
C
AC
A
C
at
K
K
K
and
L
L
L
, respectively. Let the perpendicular from
T
T
T
to
A
D
AD
A
D
meet
A
B
AB
A
B
and
A
C
AC
A
C
at
M
M
M
and
N
N
N
, respectively. Show that
∣
K
M
∣
⋅
∣
L
N
∣
=
∣
B
M
∣
⋅
∣
C
N
∣
|KM|\cdot |LN|=|BM|\cdot|CN|
∣
K
M
∣
⋅
∣
L
N
∣
=
∣
BM
∣
⋅
∣
CN
∣
.
19x19 block of 2013x2013 board has at least 21 marked sqrs
We put pebbles on some unit squares of a
2013
×
2013
2013 \times 2013
2013
×
2013
chessboard such that every unit square contains at most one pebble. Determine the minimum number of pebbles on the chessboard, if each
19
×
19
19\times 19
19
×
19
square formed by unit squares contains at least
21
21
21
pebbles.
f(x^2) = f(x)^2 - 2xf(x), f(x)=f(x-1), 1<x<y ==> f(x)<f(y)
Determine all functions
f
:
R
→
R
+
f:\mathbf{R} \rightarrow \mathbf{R}^+
f
:
R
→
R
+
such that for all real numbers
x
,
y
x,y
x
,
y
the following conditions hold:
i
.
f
(
x
2
)
=
f
(
x
)
2
−
2
x
f
(
x
)
i
i
.
f
(
−
x
)
=
f
(
x
−
1
)
i
i
i
.
1
<
x
<
y
⟹
f
(
x
)
<
f
(
y
)
.
\begin{array}{rl} i. & f(x^2) = f(x)^2 -2xf(x) \\ ii. & f(-x) = f(x-1)\\ iii. & 1<x<y \Longrightarrow f(x) < f(y). \end{array}
i
.
ii
.
iii
.
f
(
x
2
)
=
f
(
x
)
2
−
2
x
f
(
x
)
f
(
−
x
)
=
f
(
x
−
1
)
1
<
x
<
y
⟹
f
(
x
)
<
f
(
y
)
.
3
3
Hide problems
AO/OD - BC/AT=4
Let
O
O
O
be the circumcenter and
I
I
I
be the incenter of an acute triangle
A
B
C
ABC
A
BC
with
m
(
B
^
)
≠
m
(
C
^
)
m(\widehat{B}) \neq m(\widehat{C})
m
(
B
)
=
m
(
C
)
. Let
D
D
D
,
E
E
E
,
F
F
F
be the midpoints of the sides
[
B
C
]
[BC]
[
BC
]
,
[
C
A
]
[CA]
[
C
A
]
,
[
A
B
]
[AB]
[
A
B
]
, respectively. Let
T
T
T
be the foot of perpendicular from
I
I
I
to
[
A
B
]
[AB]
[
A
B
]
. Let
P
P
P
be the circumcenter of the triangle
D
E
F
DEF
D
EF
and
Q
Q
Q
be the midpoint of
[
O
I
]
[OI]
[
O
I
]
. If
A
A
A
,
P
P
P
,
Q
Q
Q
are collinear, prove that
∣
A
O
∣
∣
O
D
∣
−
∣
B
C
∣
∣
A
T
∣
=
4.
\dfrac{|AO|}{|OD|}-\dfrac{|BC|}{|AT|}=4.
∣
O
D
∣
∣
A
O
∣
−
∣
A
T
∣
∣
BC
∣
=
4.
z(xz+yz+y)/(xy+y^2+z^2+1) <= K
For all real numbers
x
,
y
,
z
x,y,z
x
,
y
,
z
such that
−
2
≤
x
,
y
,
z
≤
2
-2\leq x,y,z \leq 2
−
2
≤
x
,
y
,
z
≤
2
and
x
2
+
y
2
+
z
2
+
x
y
z
=
4
x^2+y^2+z^2+xyz = 4
x
2
+
y
2
+
z
2
+
x
yz
=
4
, determine the least real number
K
K
K
satisfying
z
(
x
z
+
y
z
+
y
)
x
y
+
y
2
+
z
2
+
1
≤
K
.
\dfrac{z(xz+yz+y)}{xy+y^2+z^2+1} \leq K.
x
y
+
y
2
+
z
2
+
1
z
(
x
z
+
yz
+
y
)
≤
K
.
Flights distributed to airway companies
Some cities of a country consisting of
n
n
n
cities are connected by round trip flights so that there are at least
k
k
k
flights from any city and any city is reachable from any city. Prove that for any such flight organization these flights can be distributed among
n
−
k
n-k
n
−
k
air companies so that one can reach any city from any city by using of at most one flight of each air company.