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Contests
National and Regional Contests
Turkey Contests
Turkey MO (2nd round)
2006 Turkey MO (2nd round)
2006 Turkey MO (2nd round)
Part of
Turkey MO (2nd round)
Subcontests
(3)
3
2
Hide problems
Find all n for which the coefficients are all divisible by 7
Find all positive integers
n
n
n
for which all coefficients of polynomial
P
(
x
)
P(x)
P
(
x
)
are divisible by
7
,
7,
7
,
where
P
(
x
)
=
(
x
2
+
x
+
1
)
n
−
(
x
2
+
1
)
n
−
(
x
+
1
)
n
−
(
x
2
+
x
)
n
+
x
2
n
+
x
n
+
1.
P(x) = (x^2 + x + 1)^n - (x^2 + 1)^n - (x + 1)^n - (x^2 + x)^n + x^{2n} + x^n + 1.
P
(
x
)
=
(
x
2
+
x
+
1
)
n
−
(
x
2
+
1
)
n
−
(
x
+
1
)
n
−
(
x
2
+
x
)
n
+
x
2
n
+
x
n
+
1.
turkey nmo 2006 q6
Find all the triangles such that its side lenghts, area and its angles' measures (in degrees) are rational.
2
2
Hide problems
2006 students and 14 teachers - find maximum of t
There are
2006
2006
2006
students and
14
14
14
teachers in a school. Each student knows at least one teacher (knowing is a symmetric relation). Suppose that, for each pair of a student and a teacher who know each other, the ratio of the number of the students whom the teacher knows to that of the teachers whom the student knows is at least
t
.
t.
t
.
Find the maximum possible value of
t
.
t.
t
.
turkey nmo 2006 q5
A
B
C
ABC
A
BC
be a triangle. Its incircle touches the sides
C
B
,
A
C
,
A
B
CB, AC, AB
CB
,
A
C
,
A
B
respectively at
N
A
,
N
B
,
N
C
N_{A},N_{B},N_{C}
N
A
,
N
B
,
N
C
. The orthic triangle of
A
B
C
ABC
A
BC
is
H
A
H
B
H
C
H_{A}H_{B}H_{C}
H
A
H
B
H
C
with
H
A
,
H
B
,
H
C
H_{A}, H_{B}, H_{C}
H
A
,
H
B
,
H
C
are respectively on
B
C
,
A
C
,
A
B
BC, AC, AB
BC
,
A
C
,
A
B
. The incenter of
A
H
C
H
B
AH_{C}H_{B}
A
H
C
H
B
is
I
A
I_{A}
I
A
;
I
B
I_{B}
I
B
and
I
C
I_{C}
I
C
were defined similarly. Prove that the hexagon
I
A
N
B
I
C
N
A
I
B
N
C
I_{A}N_{B}I_{C}N_{A}I_{B}N_{C}
I
A
N
B
I
C
N
A
I
B
N
C
has all sides equal.
1
2
Hide problems
Show that D, C, K, L lie on a circle
Points
P
P
P
and
Q
Q
Q
on side
A
B
AB
A
B
of a convex quadrilateral
A
B
C
D
ABCD
A
BC
D
are given such that
A
P
=
B
Q
.
AP = BQ.
A
P
=
BQ
.
The circumcircles of triangles
A
P
D
APD
A
P
D
and
B
Q
D
BQD
BQ
D
meet again at
K
K
K
and those of
A
P
C
APC
A
PC
and
B
Q
C
BQC
BQC
meet again at
L
L
L
. Show that the points
D
,
C
,
K
,
L
D,C,K,L
D
,
C
,
K
,
L
lie on a circle.
turkey nmo 2006 q4
x
1
,
.
.
.
,
x
n
x_{1},...,x_{n}
x
1
,
...
,
x
n
are positive reals such that their sum and their squares' sum are equal to
t
t
t
. Prove that
∑
i
≠
j
x
i
x
j
≥
(
n
−
1
)
2
⋅
t
t
−
1
\sum_{i\neq{j}}\frac{x_{i}}{x_{j}}\ge\frac{(n-1)^{2}\cdot{t}}{t-1}
∑
i
=
j
x
j
x
i
≥
t
−
1
(
n
−
1
)
2
⋅
t