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All-Russian Olympiad
1974 All Soviet Union Mathematical Olympiad
194
ASU 194 All Soviet Union MO 74 |ax+by+cz|+|bx+cy+az|+|cx+ay+bz|=|x|+|y|+|z|
ASU 194 All Soviet Union MO 74 |ax+by+cz|+|bx+cy+az|+|cx+ay+bz|=|x|+|y|+|z|
Source:
July 4, 2019
algebra
equation
Problem Statement
Find all the real
a
,
b
,
c
a,b,c
a
,
b
,
c
such that the equality
∣
a
x
+
b
y
+
c
z
∣
+
∣
b
x
+
c
y
+
a
z
∣
+
∣
c
x
+
a
y
+
b
z
∣
=
∣
x
∣
+
∣
y
∣
+
∣
z
∣
|ax+by+cz| + |bx+cy+az| + |cx+ay+bz| = |x|+|y|+|z|
∣
a
x
+
b
y
+
cz
∣
+
∣
b
x
+
cy
+
a
z
∣
+
∣
c
x
+
a
y
+
b
z
∣
=
∣
x
∣
+
∣
y
∣
+
∣
z
∣
is valid for all the real
x
,
y
,
z
x,y,z
x
,
y
,
z
.
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