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Problems
Contests
National and Regional Contests
Turkey Contests
Turkey Team Selection Test
1997 Turkey Team Selection Test
1997 Turkey Team Selection Test
Part of
Turkey Team Selection Test
Subcontests
(3)
3
2
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What is the maximal amount?
In a football league, whenever a player is transferred from a team
X
X
X
with
x
x
x
players to a team
Y
Y
Y
with
y
y
y
players, the federation is paid
y
−
x
y-x
y
−
x
billions liras by
Y
Y
Y
if
y
≥
x
y \geq x
y
≥
x
, while the federation pays
x
−
y
x-y
x
−
y
billions liras to
X
X
X
if
x
>
y
x > y
x
>
y
. A player is allowed to change as many teams as he wishes during a season. Suppose that a season started with
18
18
18
teams of
20
20
20
players each. At the end of the season,
12
12
12
of the teams turn out to have again
20
20
20
players, while the remaining
6
6
6
teams end up with
16
,
16
,
21
,
22
,
22
,
23
16,16, 21, 22, 22, 23
16
,
16
,
21
,
22
,
22
,
23
players, respectively. What is the maximal amount the federation may have won during the season?
Turkey TST 1997 Problem 6, minimum value of sum
If
x
1
,
x
2
,
…
,
x
n
x_{1}, x_{2},\ldots ,x_{n}
x
1
,
x
2
,
…
,
x
n
are positive real numbers with
x
1
2
+
x
2
2
+
…
+
x
n
2
=
1
x_{1}^2+x_2^{2}+\ldots +x_{n}^{2}=1
x
1
2
+
x
2
2
+
…
+
x
n
2
=
1
, find the minimum value of
∑
i
=
1
n
x
i
5
x
1
+
x
2
+
…
+
x
n
−
x
i
\sum_{i=1}^{n}\frac{x_{i}^{5}}{x_{1}+x_{2}+\ldots +x_{n}-x_{i}}
∑
i
=
1
n
x
1
+
x
2
+
…
+
x
n
−
x
i
x
i
5
.
2
2
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Turkey TST 1997 Problem 2, how many pairs
The sequences
(
a
n
)
(a_{n})
(
a
n
)
,
(
b
n
)
(b_{n})
(
b
n
)
are defined by
a
1
=
α
a_{1} = \alpha
a
1
=
α
,
b
1
=
β
b_{1} = \beta
b
1
=
β
,
a
n
+
1
=
α
a
n
−
β
b
n
a_{n+1} = \alpha a_{n} - \beta b_{n}
a
n
+
1
=
α
a
n
−
β
b
n
,
b
n
+
1
=
β
a
n
+
α
b
n
b_{n+1} = \beta a_{n} + \alpha b_{n}
b
n
+
1
=
β
a
n
+
α
b
n
for all
n
>
0.
n > 0.
n
>
0.
How many pairs
(
α
,
β
)
(\alpha, \beta)
(
α
,
β
)
of real numbers are there such that
a
1997
=
b
1
a_{1997} = b_{1}
a
1997
=
b
1
and
b
1997
=
a
1
b_{1997} = a_{1}
b
1997
=
a
1
?
Turkey TST 1997 Problem 5, there exist n, x_i and y_i
Show that for each prime
p
≥
7
p \geq 7
p
≥
7
, there exist a positive integer
n
n
n
and integers
x
i
x_{i}
x
i
,
y
i
y_{i}
y
i
(
i
=
1
,
.
.
.
,
n
)
(i = 1, . . . , n)
(
i
=
1
,
...
,
n
)
, not divisible by
p
p
p
, such that
x
i
2
+
y
i
2
≡
x
i
+
1
2
(
m
o
d
p
)
x_{i}^{2}+ y_{i}^{2}\equiv x_{i+1}^{2}\pmod{p}
x
i
2
+
y
i
2
≡
x
i
+
1
2
(
mod
p
)
where
x
n
+
1
=
x
1
x_{n+1} = x_{1}
x
n
+
1
=
x
1
1
2
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Turkey TST 1997 Problem 1, sum of inradii
In a triangle
A
B
C
ABC
A
BC
with a right angle at
A
A
A
,
H
H
H
is the foot of the altitude from
A
A
A
. Prove that the sum of the inradii of the triangles
A
B
C
ABC
A
BC
,
A
B
H
ABH
A
B
H
, and
A
H
C
AHC
A
H
C
is equal to
A
H
AH
A
H
.
Turkey TST 1997 Problem 4, compute AC+CE
A convex
A
B
C
D
E
ABCDE
A
BC
D
E
is inscribed in a unit circle,
A
E
AE
A
E
being its diameter. If
A
B
=
a
AB = a
A
B
=
a
,
B
C
=
b
BC = b
BC
=
b
,
C
D
=
c
CD = c
C
D
=
c
,
D
E
=
d
DE = d
D
E
=
d
and
a
b
=
c
d
=
1
4
ab = cd =\frac{1}{4}
ab
=
c
d
=
4
1
, compute
A
C
+
C
E
AC + CE
A
C
+
CE
in terms of
a
,
b
,
c
,
d
.
a, b, c, d.
a
,
b
,
c
,
d
.