MathDB
Problems
Contests
National and Regional Contests
Turkey Contests
Turkey MO (2nd round)
2007 Turkey MO (2nd round)
2007 Turkey MO (2nd round)
Part of
Turkey MO (2nd round)
Subcontests
(3)
3
2
Hide problems
Turkey NMO 2007 Problem 3, inequality with a+b+c=3
If
a
,
b
,
c
a,b,c
a
,
b
,
c
are three positive real numbers such that
a
+
b
+
c
=
3
a+b+c=3
a
+
b
+
c
=
3
, prove that
a
2
+
3
b
2
a
b
2
(
4
−
a
b
)
+
b
2
+
3
c
2
b
c
2
(
4
−
a
b
)
+
c
2
+
3
a
2
c
a
2
(
4
−
c
a
)
≥
4
{\frac{a^{2}+3b^{2}}{ab^{2}(4-ab)}}+{\frac{b^{2}+3c^{2}}{bc^{2}(4-ab)}}+{\frac{c^{2}+3a^{2}}{ca^{2}(4-ca)}}\geq 4
a
b
2
(
4
−
ab
)
a
2
+
3
b
2
+
b
c
2
(
4
−
ab
)
b
2
+
3
c
2
+
c
a
2
(
4
−
c
a
)
c
2
+
3
a
2
≥
4
Turkey NMO 2007 Problem 6, k-directionally connected
In a country between each pair of cities there is at most one direct road. There is a connection (using one or more roads) between any two cities even after the elimination of any given city and all roads incident to this city. We say that the city
A
A
A
can be k -directionally connected to the city
B
B
B
, if : we can orient at most
k
k
k
roads such that after arbitrary orientation of remaining roads for any fixed road
l
l
l
(directly connecting two cities) there is a path passing through roads in the direction of their orientation starting at
A
A
A
, passing through
l
l
l
and ending at
B
B
B
and visiting each city at most once. Suppose that in a country with
n
n
n
cities, any two cities can be k - directionally connected. What is the minimal value of
k
k
k
?
1
2
Hide problems
Turkey NMO 2007 Problem 1, m(KNB)=m(BNL)
In an acute triangle
A
B
C
ABC
A
BC
, the circle with diameter
A
C
AC
A
C
intersects
A
B
AB
A
B
and
A
C
AC
A
C
at
K
K
K
and
L
L
L
different from
A
A
A
and
C
C
C
respectively. The circumcircle of
A
B
C
ABC
A
BC
intersects the line
C
K
CK
C
K
at the point
F
F
F
different from
C
C
C
and the line
A
L
AL
A
L
at the point
D
D
D
different from
A
A
A
. A point
E
E
E
is choosen on the smaller arc of
A
C
AC
A
C
of the circumcircle of
A
B
C
ABC
A
BC
. Let
N
N
N
be the intersection of the lines
B
E
BE
BE
and
A
C
AC
A
C
. If
A
F
2
+
B
D
2
+
C
E
2
=
A
E
2
+
C
D
2
+
B
F
2
AF^{2}+BD^{2}+CE^{2}=AE^{2}+CD^{2}+BF^{2}
A
F
2
+
B
D
2
+
C
E
2
=
A
E
2
+
C
D
2
+
B
F
2
prove that
∠
K
N
B
=
∠
B
N
L
\angle KNB= \angle BNL
∠
K
NB
=
∠
BN
L
.
Turkey NMO 2007 Problem 4, (2^(m-1)-1)/127m is an integer
Let
k
>
1
k>1
k
>
1
be an integer,
p
=
6
k
+
1
p=6k+1
p
=
6
k
+
1
be a prime number and
m
=
2
p
−
1
m=2^{p}-1
m
=
2
p
−
1
.Prove that
2
m
−
1
−
1
127
m
\frac{2^{m-1}-1}{127m}
127
m
2
m
−
1
−
1
is an integer.
2
2
Hide problems
Turkey NMO 2007 Problem 2, Colored Cells on 2007x2007 board
Some unit squares of
2007
×
2007
2007\times 2007
2007
×
2007
square board are colored. Let
(
i
,
j
)
(i,j)
(
i
,
j
)
be a unit square belonging to the
i
t
h
ith
i
t
h
line and
j
t
h
jth
j
t
h
column and
S
i
,
j
S_{i,j}
S
i
,
j
be the set of all colored unit squares
(
x
,
y
)
(x,y)
(
x
,
y
)
satisfying
x
≤
i
,
y
≤
j
x\leq i, y\leq j
x
≤
i
,
y
≤
j
. At the first step in each colored unit square
(
i
,
j
)
(i,j)
(
i
,
j
)
we write the number of colored unit squares in
S
i
,
j
S_{i,j}
S
i
,
j
. In each step, in each colored unit square
(
i
,
j
)
(i,j)
(
i
,
j
)
we write the sum of all numbers written in
S
i
,
j
S_{i,j}
S
i
,
j
in the previous step. Prove that after finite number of steps, all numbers in the colored unit squares will be odd.
Turkey NMO 2007 Problem 5, CY=2MK
Let
A
B
C
ABC
A
BC
be a triangle with
∠
B
=
90
\angle B=90
∠
B
=
90
. The incircle of
A
B
C
ABC
A
BC
touches the side
B
C
BC
BC
at
D
D
D
. The incenters of triangles
A
B
D
ABD
A
B
D
and
A
D
C
ADC
A
D
C
are
X
X
X
and
Z
Z
Z
, respectively. The lines
X
Z
XZ
XZ
and
A
D
AD
A
D
are intersecting at the point
K
K
K
.
X
Z
XZ
XZ
and circumcircle of
A
B
C
ABC
A
BC
are intersecting at
U
U
U
and
V
V
V
. Let
M
M
M
be the midpoint of line segment
[
U
V
]
[UV]
[
U
V
]
.
A
D
AD
A
D
intersects the circumcircle of
A
B
C
ABC
A
BC
at
Y
Y
Y
other than
A
A
A
. Prove that
∣
C
Y
∣
=
2
∣
M
K
∣
|CY|=2|MK|
∣
C
Y
∣
=
2∣
M
K
∣
.