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XES Mathematics Olympiad
the 11th XMO
10
complex number problem, don't discuss until Tuesday
complex number problem, don't discuss until Tuesday
Source: 11th XMO Test 1 P10
February 12, 2023
complex numbers
algebra
Problem Statement
Given
t
∈
C
t\in\mathbb C
t
∈
C
. Complex numbers
x
,
y
,
z
x,y,z
x
,
y
,
z
satisfy that
∣
x
∣
=
∣
y
∣
=
∣
z
∣
=
1
|x|=|y|=|z|=1
∣
x
∣
=
∣
y
∣
=
∣
z
∣
=
1
and
t
y
=
1
x
+
1
z
\frac{t}{y}=\frac{1}{x}+\frac{1}{z}
y
t
=
x
1
+
z
1
. Calculate
∣
2
x
y
+
2
y
z
+
3
z
x
x
+
y
+
z
∣
.
\left|\frac{2xy+2yz+3zx}{x+y+z}\right|.
x
+
y
+
z
2
x
y
+
2
yz
+
3
z
x
.
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