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Contests
National and Regional Contests
Turkey Contests
Turkey Team Selection Test
1996 Turkey Team Selection Test
1996 Turkey Team Selection Test
Part of
Turkey Team Selection Test
Subcontests
(3)
3
2
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Turkey TST 1996 Problem 3
If
0
=
x
1
<
x
2
<
.
.
.
<
x
2
n
+
1
=
1
0=x_{1}<x_{2}<...<x_{2n+1}=1
0
=
x
1
<
x
2
<
...
<
x
2
n
+
1
=
1
are real numbers with
x
i
+
1
−
x
i
≤
h
x_{i+1}-x_{i} \leq h
x
i
+
1
−
x
i
≤
h
for
1
≤
i
≤
2
n
1 \leq i \leq 2n
1
≤
i
≤
2
n
, show that
1
−
h
2
<
∑
i
=
1
n
x
2
i
(
x
2
i
+
1
−
x
2
i
−
1
)
≤
1
+
h
2
\frac{1-h}{2}<\sum_{i=1}^{n}{x_{2i}(x_{2i+1}-x_{2i-1})}\leq \frac{1+h}{2}
2
1
−
h
<
∑
i
=
1
n
x
2
i
(
x
2
i
+
1
−
x
2
i
−
1
)
≤
2
1
+
h
Turkey TST 1996 Problem 6, must have the limit 0
Determine all ordered pairs of positive real numbers
(
a
,
b
)
(a, b)
(
a
,
b
)
such that every sequence
(
x
n
)
(x_{n})
(
x
n
)
satisfying
lim
n
→
∞
(
a
x
n
+
1
−
b
x
n
)
=
0
\lim_{n \rightarrow \infty}{(ax_{n+1} - bx_{n})} = 0
lim
n
→
∞
(
a
x
n
+
1
−
b
x
n
)
=
0
must have
lim
n
→
∞
x
n
=
0
\lim_{n \rightarrow \infty} x_n = 0
lim
n
→
∞
x
n
=
0
.
2
2
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Turkey TST 1996 Problem 2, P,E,F collinear
In a parallelogram
A
B
C
D
ABCD
A
BC
D
with
∠
A
<
90
\angle A < 90
∠
A
<
90
, the circle with diameter
A
C
AC
A
C
intersects the lines
C
B
CB
CB
and
C
D
CD
C
D
again at
E
E
E
and
F
F
F
, and the tangent to this circle at
A
A
A
meets the line
B
D
BD
B
D
at
P
P
P
. Prove that the points
P
P
P
,
E
E
E
,
F
F
F
are collinear.
Turkey TST 1996 Problem 5, n^2+an+b
Find the maximum number of pairwise disjoint sets of the form
S
a
,
b
=
{
n
2
+
a
n
+
b
∣
n
∈
Z
}
S_{a,b} = \{n^{2}+an+b | n \in \mathbb{Z}\}
S
a
,
b
=
{
n
2
+
an
+
b
∣
n
∈
Z
}
,
a
,
b
∈
Z
a, b \in \mathbb{Z}
a
,
b
∈
Z
.
1
2
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Turkey TST 1996 Problem 1, find a_1996
Let
∏
n
=
1
1996
(
1
+
n
x
3
n
)
=
1
+
a
1
x
k
1
+
a
2
x
k
2
+
.
.
.
+
a
m
x
k
m
\prod_{n=1}^{1996}{(1+nx^{3^n})}= 1+ a_{1}x^{k_{1}}+ a_{2}x^{k_{2}}+...+ a_{m}x^{k_{m}}
∏
n
=
1
1996
(
1
+
n
x
3
n
)
=
1
+
a
1
x
k
1
+
a
2
x
k
2
+
...
+
a
m
x
k
m
where
a
1
,
a
1
,
.
.
.
,
a
m
a_{1}, a_{1}, . . . , a_{m}
a
1
,
a
1
,
...
,
a
m
are nonzero and
k
1
<
k
2
<
.
.
.
<
k
m
k_{1} < k_{2} <...< k_{m}
k
1
<
k
2
<
...
<
k
m
. Find
a
1996
a_{1996}
a
1996
.
Turkey TST 1996 Problem 4, compute the ratio S_KLMN/S_ABC
The diagonals
A
C
AC
A
C
and
B
D
BD
B
D
of a convex quadrilateral
A
B
C
D
ABCD
A
BC
D
with
S
A
B
C
=
S
A
D
C
S_{ABC} = S_{ADC}
S
A
BC
=
S
A
D
C
intersect at
E
E
E
. The lines through
E
E
E
parallel to
A
D
AD
A
D
,
D
C
DC
D
C
,
C
B
CB
CB
,
B
A
BA
B
A
meet
A
B
AB
A
B
,
B
C
BC
BC
,
C
D
CD
C
D
,
D
A
DA
D
A
at
K
K
K
,
L
L
L
,
M
M
M
,
N
N
N
, respectively. Compute the ratio
S
K
L
M
N
S
A
B
C
\frac{S_{KLMN}}{S_{ABC}}
S
A
BC
S
K
L
MN