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National and Regional Contests
Turkey Contests
Turkey Team Selection Test
2009 Turkey Team Selection Test
2009 Turkey Team Selection Test
Part of
Turkey Team Selection Test
Subcontests
(3)
3
1
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Every student is sitting next to a friend on both sides
In a class of
n
≥
4
n\geq 4
n
≥
4
some students are friends. In this class any n \minus{} 1 students can be seated in a round table such that every student is sitting next to a friend of him in both sides, but
n
n
n
students can not be seated in that way. Prove that the minimum value of
n
n
n
is
10
10
10
.
2
2
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Turkey TST Triangle Inequality
In a triangle
A
B
C
ABC
A
BC
incircle touches the sides
A
B
AB
A
B
,
A
C
AC
A
C
and
B
C
BC
BC
at
C
1
C_1
C
1
,
B
1
B_1
B
1
and
A
1
A_1
A
1
respectively. Prove that \sqrt {\frac {AB_1}{AB}} \plus{} \sqrt {\frac {BC_1}{BC}} \plus{} \sqrt {\frac {CA_1}{CA}}\leq\frac {3}{\sqrt {2}} is true.
Turkey TST 2009 Quadrilateral
Quadrilateral
A
B
C
D
ABCD
A
BC
D
has an inscribed circle which centered at
O
O
O
with radius
r
r
r
.
A
B
AB
A
B
intersects
C
D
CD
C
D
at
P
P
P
;
A
D
AD
A
D
intersects
B
C
BC
BC
at
Q
Q
Q
and the diagonals
A
C
AC
A
C
and
B
D
BD
B
D
intersects each other at
K
K
K
. If the distance from
O
O
O
to the line
P
Q
PQ
PQ
is
k
k
k
, prove that OK\cdot\ k \equal{} r^2.
1
2
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Turkey TST 2009 Functional Equation
Find all f: Q^ \plus{} \to\ Z functions that satisfy f \left(\frac {1}{x} \right) \equal{} f(x) and (x \plus{} 1)f(x \minus{} 1) \equal{} xf(x) for all rational numbers that are bigger than 1.
Turkey TST polynominal with integer coefficients
For which
p
p
p
prime numbers, there is an integer root of the polynominal 1 \plus{} p \plus{} Q(x^1)\cdot\ Q(x^2)\ldots\ Q(x^{2p \minus{} 2}) such that
Q
(
x
)
Q(x)
Q
(
x
)
is a polynominal with integer coefficients?