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Problems
Contests
National and Regional Contests
Turkey Contests
Akdeniz University MO
1997 Akdeniz University MO
1997 Akdeniz University MO
Part of
Akdeniz University MO
Subcontests
(5)
5
2
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geometric ineq.
A
A
B
C
ABC
A
BC
triangle divide by a
d
d
d
line such that, new two pieces' areas are equal.
d
d
d
line intersects with
[
A
B
]
[AB]
[
A
B
]
at
D
D
D
,
[
A
C
]
[AC]
[
A
C
]
at
E
E
E
. Prove that
A
D
+
A
E
B
D
+
D
E
+
E
C
+
C
B
>
1
4
\frac{AD+AE}{BD+DE+EC+CB} > \frac{1}{4}
B
D
+
D
E
+
EC
+
CB
A
D
+
A
E
>
4
1
geometry
An
A
B
C
ABC
A
BC
triangle divide by a
d
d
d
line such that, new two pieces' areas and perimeters are equal. Prove that
A
B
C
ABC
A
BC
's incenter lies
d
d
d
4
2
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divide a plane
A plane dividing like a chessboard and write a real number each square such that, for a squares' number equal to its up, down ,left and right squares' numbers arithmetic mean. Prove that all number are equal.
Polygon numbers
A polygon with
1997
1997
1997
vertices is given. Write a positive real number each vertex such that, each number equal to its right and left numbers' arithmetic or geometric mean. Prove that all numbers are equal.
3
2
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sequence
(
x
n
)
(x_n)
(
x
n
)
be a sequence with
x
1
=
0
x_1=0
x
1
=
0
,
x
n
+
1
=
5
x
n
+
24
x
n
2
+
1
x_{n+1}=5x_n + \sqrt{24x_n^2+1}
x
n
+
1
=
5
x
n
+
24
x
n
2
+
1
. Prove that for
k
≥
2
k \geq 2
k
≥
2
x
k
x_k
x
k
is a natural number.
9 $\mid n$ :D
Let for all
k
∈
N
k \in {\mathbb N}
k
∈
N
k
k
k
's sum of the digits is
T
(
k
)
T(k)
T
(
k
)
. If a natural number
n
n
n
such that
T
(
n
)
=
T
(
1997
n
)
T(n)=T(1997n)
T
(
n
)
=
T
(
1997
n
)
, prove that
9
∣
n
9\mid n
9
∣
n
2
2
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Very easy inequality
If
x
x
x
and
y
y
y
are positive reals, prove that
x
2
x
y
+
y
2
y
x
≥
x
2
+
y
2
x^2\sqrt{\frac{x}{y}}+y^2\sqrt{\frac{y}{x}} \geq x^2+y^2
x
2
y
x
+
y
2
x
y
≥
x
2
+
y
2
easy inequality
Let
x
,
y
,
z
,
t
x,y,z,t
x
,
y
,
z
,
t
be real numbers such that,
1
≤
x
≤
y
≤
z
≤
t
≤
100
1 \leq x \leq y \leq z \leq t \leq 100
1
≤
x
≤
y
≤
z
≤
t
≤
100
. Find minimum value of
x
y
+
z
t
\frac{x}{y}+\frac{z}{t}
y
x
+
t
z
1
2
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Number theory
Let
m
∈
R
m \in {\mathbb R}
m
∈
R
and
x
2
+
(
m
−
4
)
x
+
(
m
2
−
3
m
+
3
)
=
0
x^2+(m-4)x+(m^2-3m+3)=0
x
2
+
(
m
−
4
)
x
+
(
m
2
−
3
m
+
3
)
=
0
equations roots are
x
1
x_1
x
1
and
x
2
x_2
x
2
and
x
1
2
+
x
2
2
=
6
x_1^2+x_2^2=6
x
1
2
+
x
2
2
=
6
. Find all
m
m
m
values.
equation
Prove that,
15
x
2
−
7
y
2
=
9
15x^2-7y^2=9
15
x
2
−
7
y
2
=
9
equation has any solutions in integers.