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Contests
National and Regional Contests
Turkey Contests
Turkey Team Selection Test
1989 Turkey Team Selection Test
1989 Turkey Team Selection Test
Part of
Turkey Team Selection Test
Subcontests
(6)
6
1
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Excenter of isosceles triangle
The circle, which is tangent to the circumcircle of isosceles triangle
A
B
C
ABC
A
BC
(
A
B
=
A
C
AB=AC
A
B
=
A
C
), is tangent
A
B
AB
A
B
and
A
C
AC
A
C
at
P
P
P
and
Q
Q
Q
, respectively. Prove that the midpoint
I
I
I
of the segment
P
Q
PQ
PQ
is the center of the excircle (which is tangent to
B
C
BC
BC
) of the triangle .
5
1
Hide problems
n weights weigh < 2n
There are
n
≥
2
n\geq2
n
≥
2
weights such that each weighs a positive integer less than
n
n
n
and their total weights is less than
2
n
2n
2
n
. Prove that there is a subset of these weights such that their total weights is equal to
n
n
n
.
4
1
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n^2 stones on nxn chessboard
There is a stone on each square of
n
×
n
n\times n
n
×
n
chessboard. We gather
n
2
n^2
n
2
stones and distribute them to the squares (again each square contains one stone) such that any two adjacent stones are again adjacent. Find all distributions such that at least one stone at the corners remains at its initial square. (Two squares are adjacent if they share a common edge.)
3
1
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Two parallel chords and a locus problem
Let
C
1
C_1
C
1
and
C
2
C_2
C
2
be given circles. Let
A
1
A_1
A
1
on
C
1
C_1
C
1
and
A
2
A_2
A
2
on
C
2
C_2
C
2
be fixed points. If chord
A
1
P
1
A_1P_1
A
1
P
1
of
C
1
C_1
C
1
is parallel to chord
A
2
P
2
A_2P_2
A
2
P
2
of
C
2
C_2
C
2
, find the locus of the midpoint of
P
1
P
2
P_1P_2
P
1
P
2
.
2
1
Hide problems
Double Numbers
A positive integer is called a "double number" if its decimal representation consists of a block of digits, not commencing with
0
0
0
, followed immediately by an identical block. So, for instance,
360360
360360
360360
is a double number, but
36036
36036
36036
is not. Show that there are infinitely many double numbers which are perfect squares.
1
1
Hide problems
f(m, m+k) = f(m,k)
Let
Z
+
\mathbb{Z}^+
Z
+
denote the set of positive integers. Find all functions
f
:
Z
+
×
Z
+
→
Z
+
f: \mathbb{Z}^+ \times \mathbb{Z}^+ \rightarrow \mathbb{Z}^+
f
:
Z
+
×
Z
+
→
Z
+
such that [*]
f
(
m
,
m
)
=
m
f(m,m)=m
f
(
m
,
m
)
=
m
[*]
f
(
m
,
k
)
=
f
(
k
,
m
)
f(m,k) = f(k,m)
f
(
m
,
k
)
=
f
(
k
,
m
)
[*]
f
(
m
,
m
+
k
)
=
f
(
m
,
k
)
f(m, m+k) = f(m,k)
f
(
m
,
m
+
k
)
=
f
(
m
,
k
)
, for each
m
,
k
∈
Z
+
m,k \in \mathbb{Z}^+
m
,
k
∈
Z
+
.