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Problems
Contests
National and Regional Contests
Russia Contests
Moscow Mathematical Olympiad
1952 Moscow Mathematical Olympiad
209
209
Part of
1952 Moscow Mathematical Olympiad
Problems
(1)
MMO 209 Moscow MO 1952 (ax+by+cz)^2+(bx+cy+az)^2+(cx+ay+bz)^2=..
Source:
8/8/2019
Prove the identity:a)
(
a
x
+
b
y
+
c
z
)
2
+
(
b
x
+
c
y
+
a
z
)
2
+
(
c
x
+
a
y
+
b
z
)
2
=
(
c
x
+
b
y
+
a
z
)
2
+
(
b
x
+
a
y
+
c
z
)
2
+
(
a
x
+
c
y
+
b
z
)
2
(ax + by + cz)^2 + (bx + cy + az)^2 + (cx + ay + bz)^2 =(cx + by + az)^2 + (bx + ay + cz)^2 + (ax + cy + bz)^2
(
a
x
+
b
y
+
cz
)
2
+
(
b
x
+
cy
+
a
z
)
2
+
(
c
x
+
a
y
+
b
z
)
2
=
(
c
x
+
b
y
+
a
z
)
2
+
(
b
x
+
a
y
+
cz
)
2
+
(
a
x
+
cy
+
b
z
)
2
b)
(
a
x
+
b
y
+
c
z
+
d
u
)
2
+
(
b
x
+
c
y
+
d
z
+
a
u
)
2
+
(
c
x
+
d
y
+
a
z
+
b
u
)
2
+
(
d
x
+
a
y
+
b
z
+
c
u
)
2
=
(ax + by + cz + du)^2+(bx + cy + dz + au)^2 +(cx + dy + az + bu)^2 + (dx + ay + bz + cu)^2 =
(
a
x
+
b
y
+
cz
+
d
u
)
2
+
(
b
x
+
cy
+
d
z
+
a
u
)
2
+
(
c
x
+
d
y
+
a
z
+
b
u
)
2
+
(
d
x
+
a
y
+
b
z
+
c
u
)
2
=
(
d
x
+
c
y
+
b
z
+
a
u
)
2
+
(
c
x
+
b
y
+
a
z
+
d
u
)
2
+
(
b
x
+
a
y
+
d
z
+
c
u
)
2
+
(
a
x
+
d
y
+
c
z
+
b
u
)
2
(dx + cy + bz + au)^2+(cx + by + az + du)^2 +(bx + ay + dz + cu)^2 + (ax + dy + cz + bu)^2
(
d
x
+
cy
+
b
z
+
a
u
)
2
+
(
c
x
+
b
y
+
a
z
+
d
u
)
2
+
(
b
x
+
a
y
+
d
z
+
c
u
)
2
+
(
a
x
+
d
y
+
cz
+
b
u
)
2
.
algebra
identity