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Contests
National and Regional Contests
Turkey Contests
Turkey Team Selection Test
1998 Turkey Team Selection Test
1998 Turkey Team Selection Test
Part of
Turkey Team Selection Test
Subcontests
(3)
3
2
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Turkey TST 1998 Problem 3, number of functions
Let
A
=
1
,
2
,
3
,
4
,
5
A = {1, 2, 3, 4, 5}
A
=
1
,
2
,
3
,
4
,
5
. Find the number of functions
f
f
f
from the nonempty subsets of
A
A
A
to
A
A
A
, such that
f
(
B
)
∈
B
f(B) \in B
f
(
B
)
∈
B
for any
B
⊂
A
B \subset A
B
⊂
A
, and
f
(
B
∪
C
)
f(B \cup C)
f
(
B
∪
C
)
is either
f
(
B
)
f(B)
f
(
B
)
or
f
(
C
)
f(C)
f
(
C
)
for any
B
B
B
,
C
⊂
A
C \subset A
C
⊂
A
Turkey TST 1998 Problem 6, N is divisible by 13
Let
f
(
x
1
,
x
2
,
.
.
.
,
x
n
)
f(x_{1}, x_{2}, . . . , x_{n})
f
(
x
1
,
x
2
,
...
,
x
n
)
be a polynomial with integer coefficients of degree less than
n
n
n
. Prove that if
N
N
N
is the number of
n
n
n
-tuples
(
x
1
,
.
.
.
,
x
n
)
(x_{1}, . . . , x_{n})
(
x
1
,
...
,
x
n
)
with
0
≤
x
i
<
13
0 \leq x_{i} < 13
0
≤
x
i
<
13
and
f
(
x
1
,
.
.
.
,
x
n
)
=
0
(
m
o
d
13
)
f(x_{1}, . . . , x_{n}) = 0 (mod 13)
f
(
x
1
,
...
,
x
n
)
=
0
(
m
o
d
13
)
, then
N
N
N
is divisible by 13.
2
2
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Turkey TST 1998 Problem 2, how many possible values of t
Let the sequence
(
a
n
)
(a_{n})
(
a
n
)
be defined by
a
1
=
t
a_{1} = t
a
1
=
t
and
a
n
+
1
=
4
a
n
(
1
−
a
n
)
a_{n+1} = 4a_{n}(1 - a_{n})
a
n
+
1
=
4
a
n
(
1
−
a
n
)
for
n
≥
1
n \geq 1
n
≥
1
. How many possible values of t are there, if
a
1998
=
0
a_{1998} = 0
a
1998
=
0
?
Turkey TST 1998 Problem 5, AF=AC
In a triangle
A
B
C
ABC
A
BC
, the circle through
C
C
C
touching
A
B
AB
A
B
at
A
A
A
and the circle through
B
B
B
touching
A
C
AC
A
C
at
A
A
A
have different radii and meet again at
D
D
D
. Let
E
E
E
be the point on the ray
A
B
AB
A
B
such that
A
B
=
B
E
AB = BE
A
B
=
BE
. The circle through
A
A
A
,
D
D
D
,
E
E
E
intersect the ray
C
A
CA
C
A
again at
F
F
F
. Prove that
A
F
=
A
C
AF = AC
A
F
=
A
C
.
1
2
Hide problems
Turkey TST 1998 Problem 1, Squares BAXX' and CAYY'
Squares
B
A
X
X
′
BAXX^{'}
B
A
X
X
′
and
C
A
Y
Y
′
CAYY^{'}
C
A
Y
Y
′
are drawn in the exterior of a triangle
A
B
C
ABC
A
BC
with
A
B
=
A
C
AB = AC
A
B
=
A
C
. Let
D
D
D
be the midpoint of
B
C
BC
BC
, and
E
E
E
and
F
F
F
be the feet of the perpendiculars from an arbitrary point
K
K
K
on the segment
B
C
BC
BC
to
B
Y
BY
B
Y
and
C
X
CX
CX
, respectively.
(
a
)
(a)
(
a
)
Prove that
D
E
=
D
F
DE = DF
D
E
=
D
F
.
(
b
)
(b)
(
b
)
Find the locus of the midpoint of
E
F
EF
EF
.
Turkey TST 1998 Problem 4, n houses and n people
Suppose
n
n
n
houses are to be assigned to
n
n
n
people. Each person ranks the houses in the order of preference, with no ties. After the assignment is made, it is observed that every other assignment would assign to at least one person a less preferred house. Prove that there is at least one person who received the house he/she preferred most under this assignment.