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India IOQM
2022-23 IOQM India
14
IOQM 2022-23 P-14
IOQM 2022-23 P-14
Source:
October 30, 2022
algebra
complex numbers
Problem Statement
Let
x
,
y
,
z
x,y,z
x
,
y
,
z
be complex numbers such that\\
x
y
+
z
+
y
z
+
x
+
z
x
+
y
=
9
\hspace{ 2cm} \frac{x}{y+z}+\frac{y}{z+x}+\frac{z}{x+y}=9
y
+
z
x
+
z
+
x
y
+
x
+
y
z
=
9
\\
x
2
y
+
z
+
y
2
z
+
x
+
z
2
x
+
y
=
64
\hspace{ 2cm} \frac{x^2}{y+z}+\frac{y^2}{z+x}+\frac{z^2}{x+y}=64
y
+
z
x
2
+
z
+
x
y
2
+
x
+
y
z
2
=
64
\\
x
3
y
+
z
+
y
3
z
+
x
+
z
3
x
+
y
=
488
\hspace{ 2cm} \frac{x^3}{y+z}+\frac{y^3}{z+x}+\frac{z^3}{x+y}=488
y
+
z
x
3
+
z
+
x
y
3
+
x
+
y
z
3
=
488
\\ \\ If
x
y
z
+
y
z
x
+
z
x
y
=
m
n
\frac{x}{yz}+\frac{y}{zx}+\frac{z}{xy}=\frac{m}{n}
yz
x
+
z
x
y
+
x
y
z
=
n
m
where
m
,
n
m,n
m
,
n
are positive integers with
G
C
D
(
m
,
n
)
=
1
GCD(m,n)=1
GC
D
(
m
,
n
)
=
1
, find
m
+
n
m+n
m
+
n
.
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