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Problems
Contests
National and Regional Contests
Turkey Contests
Akdeniz University MO
1998 Akdeniz University MO
5
5
Part of
1998 Akdeniz University MO
Problems
(2)
system of equations
Source:
1/30/2016
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
system of equations
number theory
algebra
geomeetry
Source:
1/31/2016
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
geometry
quadrilateral