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National and Regional Contests
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MAA AMC
AMC 12/AHSME
1964 AMC 12/AHSME
32
32
Part of
1964 AMC 12/AHSME
Problems
(1)
Deduction from a fraction
Source: AHSME 1964 Problem 32
1/13/2014
If
a
+
b
b
+
c
=
c
+
d
d
+
a
\dfrac{a+b}{b+c}=\dfrac{c+d}{d+a}
b
+
c
a
+
b
=
d
+
a
c
+
d
, then:
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
a
must equal
c
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
a
+
b
+
c
+
d
must equal zero
<span class='latex-bold'>(A)</span>\ a\text{ must equal }c \qquad <span class='latex-bold'>(B)</span>\ a+b+c+d\text{ must equal zero }\qquad
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
a
must equal
c
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
a
+
b
+
c
+
d
must equal zero
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
either
a
=
c
or
a
+
b
+
c
+
d
=
0
,
or both
<span class='latex-bold'>(C)</span>\ \text{either }a=c\text{ or }a+b+c+d=0,\text{ or both} \qquad
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
either
a
=
c
or
a
+
b
+
c
+
d
=
0
,
or both
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
a
+
b
+
c
+
d
≠
0
if
a
=
c
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
a
(
b
+
c
+
d
)
=
c
(
a
+
b
+
d
)
<span class='latex-bold'>(D)</span>\ a+b+c+d\neq 0\text{ if }a=c \qquad <span class='latex-bold'>(E)</span>\ a(b+c+d)=c(a+b+d)
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
a
+
b
+
c
+
d
=
0
if
a
=
c
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
a
(
b
+
c
+
d
)
=
c
(
a
+
b
+
d
)
AMC