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Ireland National Math Olympiad
2019 Irish Math Olympiad
4
4
Part of
2019 Irish Math Olympiad
Problems
(1)
1 +1/x + 2(x + 1)/xy + 3(x + 1)(y + 2)/xyz+4(x + 1)(y + 2)(z + 3)/xyz=0
Source: 2019 Irish Mathematical Olympiad paper 1 p4
10/5/2020
Find the set of all quadruplets
(
x
,
y
,
z
,
w
)
(x,y, z,w)
(
x
,
y
,
z
,
w
)
of non-zero real numbers which satisfy
1
+
1
x
+
2
(
x
+
1
)
x
y
+
3
(
x
+
1
)
(
y
+
2
)
x
y
z
+
4
(
x
+
1
)
(
y
+
2
)
(
z
+
3
)
x
y
z
w
=
0
1 +\frac{1}{x}+\frac{2(x + 1)}{xy}+\frac{3(x + 1)(y + 2)}{xyz}+\frac{4(x + 1)(y + 2)(z + 3)}{xyzw}= 0
1
+
x
1
+
x
y
2
(
x
+
1
)
+
x
yz
3
(
x
+
1
)
(
y
+
2
)
+
x
yz
w
4
(
x
+
1
)
(
y
+
2
)
(
z
+
3
)
=
0
algebra
equation