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Problems
Contests
National and Regional Contests
Turkey Contests
Akdeniz University MO
1996 Akdeniz University MO
1996 Akdeniz University MO
Part of
Akdeniz University MO
Subcontests
(5)
5
1
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geometry
Two circles centered
O
1
,
O
2
O_1,O_2
O
1
,
O
2
intersects at two points
M
M
M
and
N
N
N
.
O
1
M
O_1M
O
1
M
line intersects with
O
1
O_1
O
1
centered circle and
O
2
O_2
O
2
centered circle at
A
1
A_1
A
1
and
A
2
A_2
A
2
,
O
2
M
O_2M
O
2
M
line intersects with
O
1
O_1
O
1
centered circle and
O
2
O_2
O
2
centered circle at
B
1
B_1
B
1
and
B
2
B_2
B
2
respectively. Let
K
K
K
is intersection point of the
A
1
B
1
A_1B_1
A
1
B
1
and
A
2
B
2
A_2B_2
A
2
B
2
. Prove that
N
,
M
,
K
N,M,K
N
,
M
,
K
collinear.
4
1
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points in a plane
25
25
25
point in a plane and for all
3
3
3
points, we find
2
2
2
points such that this
2
2
2
points' distance less than
1
1
1
c
m
cm
c
m
. Prove that at least
13
13
13
points in a circle of radius
1
1
1
c
m
cm
c
m
.
3
1
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an easy question
A
x
>
2
x>2
x
>
2
real number is given. Bob has got
1997
1997
1997
labels and writes one of the numbers
"
x
0
,
x
1
,
x
2
,
⋯
x
1995
,
x
1996
"
"x^0, x^1, x^2 ,\dotsm x^{1995}, x^{1996}"
"
x
0
,
x
1
,
x
2
,
⋯
x
1995
,
x
1996
"
each labels such that all labels has distinct numbers. Bob puts some labels to right pocket, some labels to left pocket. Prove that sum of numbers of the right pocket never equal to sum of numbers of the left pocket.
2
1
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fibonacci sequence
Let
u
1
=
1
,
u
2
=
1
u_1=1,u_2=1
u
1
=
1
,
u
2
=
1
and for all
k
≥
1
k \geq 1
k
≥
1
's
u
k
+
2
=
u
k
+
1
+
u
k
u_{k+2}=u_{k+1}+u_{k}
u
k
+
2
=
u
k
+
1
+
u
k
Prove that for all
m
≥
1
m \geq 1
m
≥
1
's
5
5
5
divides
u
5
m
u_{5m}
u
5
m
1
1
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Easy equation
Solve the equation for real numbers
x
,
y
,
z
x,y,z
x
,
y
,
z
(
x
−
y
+
z
)
2
=
x
2
−
y
2
+
z
2
(x-y+z)^2=x^2-y^2+z^2
(
x
−
y
+
z
)
2
=
x
2
−
y
2
+
z
2