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Problems
Contests
National and Regional Contests
Turkey Contests
Turkey Team Selection Test
2003 Turkey Team Selection Test
2003 Turkey Team Selection Test
Part of
Turkey Team Selection Test
Subcontests
(6)
4
1
Hide problems
(x^2+y^2)^2 + 2tx(x^2 + y^2) = t^2y^2
Find the leasta. positive real numberb. positive integer
t
t
t
such that the equation
(
x
2
+
y
2
)
2
+
2
t
x
(
x
2
+
y
2
)
=
t
2
y
2
(x^2+y^2)^2 + 2tx(x^2 + y^2) = t^2y^2
(
x
2
+
y
2
)
2
+
2
t
x
(
x
2
+
y
2
)
=
t
2
y
2
has a solution where
x
,
y
x,y
x
,
y
are positive integers.
1
1
Hide problems
M = {(a,b,c,d)|a,b,c,d € {1,2,3,4} and abcd > 1}
Let
M
=
{
(
a
,
b
,
c
,
d
)
∣
a
,
b
,
c
,
d
∈
{
1
,
2
,
3
,
4
}
and
a
b
c
d
>
1
}
M = \{(a,b,c,d)|a,b,c,d \in \{1,2,3,4\} \text{ and } abcd > 1\}
M
=
{(
a
,
b
,
c
,
d
)
∣
a
,
b
,
c
,
d
∈
{
1
,
2
,
3
,
4
}
and
ab
c
d
>
1
}
. For each
n
∈
{
1
,
2
,
…
,
254
}
n\in \{1,2,\dots, 254\}
n
∈
{
1
,
2
,
…
,
254
}
, the sequence
(
a
1
,
b
1
,
c
1
,
d
1
)
(a_1, b_1, c_1, d_1)
(
a
1
,
b
1
,
c
1
,
d
1
)
,
(
a
2
,
b
2
,
c
2
,
d
2
)
(a_2, b_2, c_2, d_2)
(
a
2
,
b
2
,
c
2
,
d
2
)
,
…
\dots
…
,
(
a
255
,
b
255
,
c
255
,
d
255
)
(a_{255}, b_{255},c_{255},d_{255})
(
a
255
,
b
255
,
c
255
,
d
255
)
contains each element of
M
M
M
exactly once and the equality
∣
a
n
+
1
−
a
n
∣
+
∣
b
n
+
1
−
b
n
∣
+
∣
c
n
+
1
−
c
n
∣
+
∣
d
n
+
1
−
d
n
∣
=
1
|a_{n+1} - a_n|+|b_{n+1} - b_n|+|c_{n+1} - c_n|+|d_{n+1} - d_n| = 1
∣
a
n
+
1
−
a
n
∣
+
∣
b
n
+
1
−
b
n
∣
+
∣
c
n
+
1
−
c
n
∣
+
∣
d
n
+
1
−
d
n
∣
=
1
holds. If
c
1
=
d
1
=
1
c_1 = d_1 = 1
c
1
=
d
1
=
1
, find all possible values of the pair
(
a
1
,
b
1
)
(a_1,b_1)
(
a
1
,
b
1
)
.
6
1
Hide problems
Non-decreasing integ seq. with sum of terms is equal to n
For all positive integers
n
n
n
, let
p
(
n
)
p(n)
p
(
n
)
be the number of non-decreasing sequences of positive integers such that for each sequence, the sum of all terms of the sequence is equal to
n
n
n
. Prove that
1
+
p
(
1
)
+
p
(
2
)
+
⋯
+
p
(
n
−
1
)
p
(
n
)
≤
2
n
.
\dfrac{1+p(1)+p(2) + \dots + p(n-1)}{p(n)} \leq \sqrt {2n}.
p
(
n
)
1
+
p
(
1
)
+
p
(
2
)
+
⋯
+
p
(
n
−
1
)
≤
2
n
.
3
1
Hide problems
Arithmetic seq with each term is power of a natural number
Is there an arithmetic sequence witha.
2003
2003
2003
b. infinitely many terms such that each term is a power of a natural number with a degree greater than
1
1
1
?
5
1
Hide problems
Reflection of midpoint of a chord over a point on diameter
Let
A
A
A
be a point on a circle with center
O
O
O
and
B
B
B
be the midpoint of
[
O
A
]
[OA]
[
O
A
]
. Let
C
C
C
and
D
D
D
be points on the circle such that they lie on the same side of the line
O
A
OA
O
A
and
C
B
O
^
=
D
B
A
^
\widehat{CBO} = \widehat{DBA}
CBO
=
D
B
A
. Show that the reflection of the midpoint of
[
C
D
]
[CD]
[
C
D
]
over
B
B
B
lies on the circle.
2
1
Hide problems
[KLMN]/[ABCD]<8/27
Let
K
K
K
be the intersection of the diagonals of a convex quadrilateral
A
B
C
D
ABCD
A
BC
D
. Let
L
∈
[
A
D
]
L\in [AD]
L
∈
[
A
D
]
,
M
∈
[
A
C
]
M \in [AC]
M
∈
[
A
C
]
,
N
∈
[
B
C
]
N \in [BC]
N
∈
[
BC
]
such that
K
L
∥
A
B
KL\parallel AB
K
L
∥
A
B
,
L
M
∥
D
C
LM\parallel DC
L
M
∥
D
C
,
M
N
∥
A
B
MN\parallel AB
MN
∥
A
B
. Show that
A
r
e
a
(
K
L
M
N
)
A
r
e
a
(
A
B
C
D
)
<
8
27
.
\dfrac{Area(KLMN)}{Area(ABCD)} < \dfrac {8}{27}.
A
re
a
(
A
BC
D
)
A
re
a
(
K
L
MN
)
<
27
8
.