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Contests
National and Regional Contests
Turkey Contests
Akdeniz University MO
1999 Akdeniz University MO
1999 Akdeniz University MO
Part of
Akdeniz University MO
Subcontests
(5)
5
2
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geometry
A circle centered with
O
O
O
.
C
C
C
is a stable point in circle. A chord
[
A
B
]
[AB]
[
A
B
]
, parallel to
O
C
OC
OC
.Prove that,
[
A
C
]
2
+
[
B
C
]
2
[AC]^2+[BC]^2
[
A
C
]
2
+
[
BC
]
2
is stable.
geometry
Let
C
C
C
is at a circle.
[
A
B
]
[AB]
[
A
B
]
is a diameter this circle.
D
D
D
is a point at
[
A
B
]
[AB]
[
A
B
]
. Perpendicular from
C
C
C
to
[
A
B
]
[AB]
[
A
B
]
's foot on the
[
A
B
]
[AB]
[
A
B
]
is
E
E
E
, perpendicular from
A
A
A
to
[
C
D
]
[CD]
[
C
D
]
's foot on the
[
C
D
]
[CD]
[
C
D
]
is
F
F
F
. Prove that,
[
D
C
]
[
F
C
]
=
[
B
D
]
[
E
A
]
[DC][FC]=[BD][EA]
[
D
C
]
[
FC
]
=
[
B
D
]
[
E
A
]
4
2
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number theory
In a sequence ,first term is
2
2
2
and after
2.
2.
2.
term all terms is equal to sum of the previous number's digits'
5.
5.
5.
power. (Like this
2.
2.
2.
term is
2
5
=
32
2^5=32
2
5
=
32
,
3.
3.
3.
term is
3
5
+
2
5
=
243
+
32
=
275
⋯
3^5+2^5=243+32=275\dotsm
3
5
+
2
5
=
243
+
32
=
275
⋯
) Prove that, this infinite sequence has at least
2
2
2
two numbers are equal.
geometric combinatorics
Placing
n
∈
N
n \in {\mathbb N}
n
∈
N
circles with radius
1
1
1
u
n
i
t
unit
u
ni
t
inside a square with side
100
100
100
u
n
i
t
unit
u
ni
t
such that, whichever line segment with lenght
10
10
10
u
n
i
t
unit
u
ni
t
intersect at least one circle. Prove that
n
≥
416
n \geq 416
n
≥
416
3
2
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inequality
Let
a
a
a
,
b
b
b
,
c
c
c
and
d
d
d
positive reals. Prove that
1
a
+
b
+
c
+
d
≤
1
64
(
1
a
+
1
b
+
4
c
+
16
d
)
\frac{1}{a+b+c+d} \leq \frac{1}{64}(\frac{1}{a}+\frac{1}{b}+\frac{4}{c}+\frac{16}{d})
a
+
b
+
c
+
d
1
≤
64
1
(
a
1
+
b
1
+
c
4
+
d
16
)
inequality
For all
x
>
2
x> \sqrt 2
x
>
2
,
y
>
2
y> \sqrt 2
y
>
2
numbers, prove that
x
4
−
x
3
y
+
x
2
y
2
−
x
y
3
+
y
4
>
x
2
+
y
2
x^4-x^3y+x^2y^2-xy^3+y^4>x^2+y^2
x
4
−
x
3
y
+
x
2
y
2
−
x
y
3
+
y
4
>
x
2
+
y
2
2
2
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number theory
Prove that, we can't find positive numbers
m
m
m
and
n
n
n
such that,
m
2
+
(
m
+
1
)
2
=
n
4
+
(
n
+
1
)
4
m^2+(m+1)^2=n^4+(n+1)^4
m
2
+
(
m
+
1
)
2
=
n
4
+
(
n
+
1
)
4
number theory
Find all
(
x
,
y
)
(x,y)
(
x
,
y
)
real numbers pairs such that,
x
7
+
y
7
=
x
4
+
y
4
x^7+y^7=x^4+y^4
x
7
+
y
7
=
x
4
+
y
4
1
2
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number theory
Prove that, we find infinite numbers such that, this number writeable
1999
k
+
1
1999k+1
1999
k
+
1
for
k
∈
N
k \in {\mathbb N}
k
∈
N
and all digits are equal.
number theory
Let
n
n
n
's positive divisors sum is
T
(
n
)
T(n)
T
(
n
)
. For all
n
≥
3
n \geq 3
n
≥
3
's prove that,
(
T
(
n
)
)
3
<
n
4
(T(n))^3<n^4
(
T
(
n
)
)
3
<
n
4