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National and Regional Contests
Turkey Contests
Turkey Junior National Olympiad
2000 Turkey Junior National Olympiad
2000 Turkey Junior National Olympiad
Part of
Turkey Junior National Olympiad
Subcontests
(3)
3
1
Hide problems
Turkey Junior Olympiad 2000, Part II - P3
f
:
R
→
R
f:\mathbb{R}\rightarrow \mathbb{R}
f
:
R
→
R
satisfies the equation
f
(
x
)
f
(
y
)
−
a
f
(
x
y
)
=
x
+
y
f(x)f(y)-af(xy)=x+y
f
(
x
)
f
(
y
)
−
a
f
(
x
y
)
=
x
+
y
, for every real numbers
x
,
y
x,y
x
,
y
. Find all possible real values of
a
a
a
.
2
1
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Turkey Junior Olympiad 2000, Part II - P2
Find the least positive integer
n
n
n
such that
15
15
15
divides the product
a
1
a
2
…
a
15
(
a
1
n
+
a
2
n
+
⋯
+
a
15
n
)
a_1a_2\dots a_{15}\left (a_1^n+a_2^n+\dots+a_{15}^n \right )
a
1
a
2
…
a
15
(
a
1
n
+
a
2
n
+
⋯
+
a
15
n
)
, for every positive integers
a
1
,
a
2
,
…
,
a
15
a_1, a_2, \dots, a_{15}
a
1
,
a
2
,
…
,
a
15
.
1
1
Hide problems
Turkey Junior Olympiad 2000, Part II - P1
Let
A
B
C
ABC
A
BC
be a triangle with
∠
B
A
C
=
9
0
∘
\angle BAC = 90^\circ
∠
B
A
C
=
9
0
∘
. Construct the square
B
D
E
C
BDEC
B
D
EC
such as
A
A
A
and the square are at opposite sides of
B
C
BC
BC
. Let the angle bisector of
∠
B
A
C
\angle BAC
∠
B
A
C
cut the sides
[
B
C
]
[BC]
[
BC
]
and
[
D
E
]
[DE]
[
D
E
]
at
F
F
F
and
G
G
G
, respectively. If
∣
A
B
∣
=
24
|AB|=24
∣
A
B
∣
=
24
and
∣
A
C
∣
=
10
|AC|=10
∣
A
C
∣
=
10
, calculate the area of quadrilateral
B
D
G
F
BDGF
B
D
GF
.