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International Mathematical Olympiad
›
2001
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Problem 6
Problem 6
Let
a
>
b
>
c
>
d
>
0
a>b>c>d>0
a
>
b
>
c
>
d
>
0
be integers satisfying
a
c
+
b
d
=
(
b
+
d
+
a
−
c
)
(
b
+
d
−
a
+
c
)
ac+bd=(b+d+a-c)(b+d-a+c)
a
c
+
b
d
=
(
b
+
d
+
a
−
c
)
(
b
+
d
−
a
+
c
)
. Prove that
a
b
+
c
d
ab+cd
ab
+
c
d
is not prime.
Step 3 of 6: Apply Ptolemy
Previous step
Next step
W
Y
2
=
(
a
b
+
c
d
)
(
a
d
+
b
c
)
a
c
+
b
d
.
WY^2=\frac{(ab+cd)(ad+bc)}{ac+bd}.
W
Y
2
=
a
c
+
b
d
(
ab
+
c
d
)
(
a
d
+
b
c
)
.
Detailed analysis
Ptolemy and the cosine-rule diagonal formulas give this identity.
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