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Problems
Contests
National and Regional Contests
Turkey Contests
Akdeniz University MO
1998 Akdeniz University MO
1998 Akdeniz University MO
Part of
Akdeniz University MO
Subcontests
(5)
5
2
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system of equations
Solve the equation system for real numbers:
x
1
+
x
2
=
x
3
2
x_1+x_2=x_3^2
x
1
+
x
2
=
x
3
2
x
2
+
x
3
=
x
4
2
x_2+x_3=x_4^2
x
2
+
x
3
=
x
4
2
x
3
+
x
4
=
x
1
2
x_3+x_4=x_1^2
x
3
+
x
4
=
x
1
2
x
4
+
x
1
=
x
2
2
x_4+x_1=x_2^2
x
4
+
x
1
=
x
2
2
geomeetry
Let
A
B
C
D
ABCD
A
BC
D
a convex quadrilateral with
[
B
C
]
[BC]
[
BC
]
and
[
C
D
]
[CD]
[
C
D
]
's midpoint is
P
P
P
and
N
N
N
respectively. If
[
A
P
]
+
[
A
N
]
=
d
[AP]+[AN]=d
[
A
P
]
+
[
A
N
]
=
d
Show that, area of the
A
B
C
D
ABCD
A
BC
D
is less then
1
2
d
2
\frac{1}{2}d^2
2
1
d
2
4
2
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Geometry
Let
A
B
C
ABC
A
BC
be an equilateral triangle with side lenght is
1
1
1
c
m
cm
c
m
.Let
D
∈
[
A
B
]
D \in [AB]
D
∈
[
A
B
]
is a point. Perpendiculars from
D
D
D
to
[
A
C
]
[AC]
[
A
C
]
and
[
B
C
]
[BC]
[
BC
]
intersects with
[
A
C
]
[AC]
[
A
C
]
and
[
B
C
]
[BC]
[
BC
]
at points
E
E
E
and
F
F
F
respectively. Perpendiculars from
E
E
E
and
F
F
F
to
[
A
B
]
[AB]
[
A
B
]
intersects with
[
A
B
]
[AB]
[
A
B
]
at points
E
1
E_1
E
1
and
F
1
F_1
F
1
. Prove that
[
E
1
F
1
]
=
3
4
[E_1F_1]=\frac{3}{4}
[
E
1
F
1
]
=
4
3
2 \times 11 to 1 \times 2
A floor has
2
×
11
2 \times 11
2
×
11
dimension, and this floor covering with
1
×
2
1 \times 2
1
×
2
rectangles. (No two rectangles overlap). How many cases we done this job?
3
2
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Inequality (easy)
Let
x
,
y
,
z
x,y,z
x
,
y
,
z
be non-negative numbers such that
x
+
y
+
z
≤
3
x+y+z \leq 3
x
+
y
+
z
≤
3
. Prove that
2
1
+
x
+
2
1
+
y
+
2
1
+
z
≥
3
\frac{2}{1+x}+\frac{2}{1+y}+\frac{2}{1+z} \geq 3
1
+
x
2
+
1
+
y
2
+
1
+
z
2
≥
3
inequality
Let
x
,
y
,
z
x,y,z
x
,
y
,
z
be real numbers such that,
x
≥
y
≥
z
>
0
x \geq y \geq z >0
x
≥
y
≥
z
>
0
. Prove that
x
2
−
y
2
z
+
z
2
−
y
2
x
+
x
2
−
z
2
y
≥
3
x
−
4
y
+
z
\frac{x^2-y^2}{z}+\frac{z^2-y^2}{x}+\frac{x^2-z^2}{y} \geq 3x-4y+z
z
x
2
−
y
2
+
x
z
2
−
y
2
+
y
x
2
−
z
2
≥
3
x
−
4
y
+
z
2
2
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Plane points...
100
100
100
points at a circle with radius
1
1
1
c
m
cm
c
m
. Show that, we find an another point such that, this point's distance to other
100
100
100
points is greater than
100
100
100
c
m
cm
c
m
.
geometry
We have
1998
1998
1998
polygon such that sum of the areas is
1997
,
5
1997,5
1997
,
5
c
m
2
cm^2
c
m
2
. These polygons placing inside a square with side lenght
1
1
1
c
m
cm
c
m
. (Polygons no overflow). Prove that we can find a point such that, all polygons have this point.
1
2
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perfect square (easy)
Prove that, for
k
∈
Z
+
k \in {\mathbb Z^+}
k
∈
Z
+
k
(
k
+
1
)
(
k
+
2
)
(
k
+
3
)
k(k+1)(k+2)(k+3)
k
(
k
+
1
)
(
k
+
2
)
(
k
+
3
)
is not a perfect square.
question
Whichever
3
3
3
odd numbers is given. Prove that we can find a
4.
4.
4.
odd number such that, sum of squares of the these numbers is a perfect square.