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Problems
Contests
National and Regional Contests
Turkey Contests
Turkey MO (2nd round)
2003 Turkey MO (2nd round)
2003 Turkey MO (2nd round)
Part of
Turkey MO (2nd round)
Subcontests
(3)
3
2
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for all real a_1 , a_2 , ... , a_2004
Let
f
:
R
→
R
f: \mathbb R \rightarrow \mathbb R
f
:
R
→
R
be a function such that f(tx_1\plus{}(1\minus{}t)x_2)\leq tf(x_1)\plus{}(1\minus{}t)f(x_2) for all
x
1
,
x
2
∈
R
x_1 , x_2 \in \mathbb R
x
1
,
x
2
∈
R
and
t
∈
(
0
,
1
)
t\in (0,1)
t
∈
(
0
,
1
)
. Show that \sum_{k\equal{}1}^{2003}f(a_{k\plus{}1})a_k \geq \sum_{k\equal{}1}^{2003}f(a_k)a_{k\plus{}1} for all real numbers
a
1
,
a
2
,
.
.
.
,
a
2004
a_1,a_2,...,a_{2004}
a
1
,
a
2
,
...
,
a
2004
such that
a
1
≥
a
2
≥
.
.
.
≥
a
2003
a_1\geq a_2\geq ... \geq a_{2003}
a
1
≥
a
2
≥
...
≥
a
2003
and a_{2004}\equal{}a_1
mxn chessboard largest beautiful number
An assignment of either a
0
0
0
or a
1
1
1
to each unit square of an
m
m
m
x
n
n
n
chessboard is called
f
a
i
r
fair
f
ai
r
if the total numbers of
0
0
0
s and
1
1
1
s are equal. A real number
a
a
a
is called
b
e
a
u
t
i
f
u
l
beautiful
b
e
a
u
t
i
f
u
l
if there are positive integers
m
,
n
m,n
m
,
n
and a fair assignment for the
m
m
m
x
n
n
n
chessboard such that for each of the
m
m
m
rows and
n
n
n
columns , the percentage of
1
1
1
s on that row or column is not less than
a
a
a
or greater than 100\minus{}a. Find the largest beautiful number.
1
2
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n cars participating in a rally
n
≥
2
n\geq 2
n
≥
2
cars are participating in a rally. The cars leave the start line at different times and arrive at the finish line at different times. During the entire rally each car takes over any other car at most once , the number of cars taken over by each car is different and each car is taken over by the same number of cars. Find all possible values of
n
n
n
Turkey NMO 2003 Problem 4
Suppose that
2
2
n
+
1
+
2
n
+
1
=
x
k
2^{2n+1}+ 2^{n}+1=x^{k}
2
2
n
+
1
+
2
n
+
1
=
x
k
, where
k
≥
2
k\geq2
k
≥
2
and
n
n
n
are positive integers. Find all possible values of
n
n
n
.
2
2
Hide problems
Geometric inequality.
Let
A
B
C
D
ABCD
A
BC
D
be a convex quadrilateral and
K
,
L
,
M
,
N
K,L,M,N
K
,
L
,
M
,
N
be points on
[
A
B
]
,
[
B
C
]
,
[
C
D
]
,
[
D
A
]
[AB],[BC],[CD],[DA]
[
A
B
]
,
[
BC
]
,
[
C
D
]
,
[
D
A
]
, respectively. Show that,
s
1
3
+
s
2
3
+
s
3
3
+
s
4
3
≤
2
s
3
\sqrt[3]{s_{1}}+\sqrt[3]{s_{2}}+\sqrt[3]{s_{3}}+\sqrt[3]{s_{4}}\leq 2\sqrt[3]{s}
3
s
1
+
3
s
2
+
3
s
3
+
3
s
4
≤
2
3
s
where
s
1
=
Area
(
A
K
N
)
s_1=\text{Area}(AKN)
s
1
=
Area
(
A
K
N
)
,
s
2
=
Area
(
B
K
L
)
s_2=\text{Area}(BKL)
s
2
=
Area
(
B
K
L
)
,
s
3
=
Area
(
C
L
M
)
s_3=\text{Area}(CLM)
s
3
=
Area
(
C
L
M
)
,
s
4
=
Area
(
D
M
N
)
s_4=\text{Area}(DMN)
s
4
=
Area
(
D
MN
)
and
s
=
Area
(
A
B
C
D
)
s=\text{Area}(ABCD)
s
=
Area
(
A
BC
D
)
.
I incenter show that ...
A circle which is tangent to the sides
[
A
B
]
[AB]
[
A
B
]
and
[
B
C
]
[BC]
[
BC
]
of
△
A
B
C
\triangle ABC
△
A
BC
is also tangent to its circumcircle at the point
T
T
T
. If
I
I
I
is the incenter of
△
A
B
C
\triangle ABC
△
A
BC
, show that \widehat{ATI}\equal{}\widehat{CTI}