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2023 ISL
N7
Hardest N7 in history
Hardest N7 in history
Source: ISL 2023 N7
July 17, 2024
number theory
Problem Statement
Let
a
,
b
,
c
,
d
a,b,c,d
a
,
b
,
c
,
d
be positive integers satisfying
a
b
a
+
b
+
c
d
c
+
d
=
(
a
+
b
)
(
c
+
d
)
a
+
b
+
c
+
d
.
\frac{ab}{a+b}+\frac{cd}{c+d}=\frac{(a+b)(c+d)}{a+b+c+d}.
a
+
b
ab
+
c
+
d
c
d
=
a
+
b
+
c
+
d
(
a
+
b
)
(
c
+
d
)
.
Determine all possible values of
a
+
b
+
c
+
d
a+b+c+d
a
+
b
+
c
+
d
.
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