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Problems
Contests
National and Regional Contests
Turkey Contests
National Olympiad First Round
2007 National Olympiad First Round
15
15
Part of
2007 National Olympiad First Round
Problems
(1)
Turkey NMO 2007 1st Round - P15 (Algebra)
Source:
10/4/2012
What is the minimum value of
a
b
+
c
d
ab+cd
ab
+
c
d
, if
a
b
+
c
d
=
e
f
+
g
h
ab+cd = ef+gh
ab
+
c
d
=
e
f
+
g
h
where
a
,
b
,
c
,
d
,
e
,
f
,
g
,
h
a,b,c,d,e,f,g,h
a
,
b
,
c
,
d
,
e
,
f
,
g
,
h
are distinct positive integers?
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
34
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
33
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
32
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
31
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
30
<span class='latex-bold'>(A)</span>\ 34 \qquad<span class='latex-bold'>(B)</span>\ 33 \qquad<span class='latex-bold'>(C)</span>\ 32 \qquad<span class='latex-bold'>(D)</span>\ 31 \qquad<span class='latex-bold'>(E)</span>\ 30
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
34
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
33
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
32
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
31
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
30