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Cyclic equality implies equal sum of squares
Cyclic equality implies equal sum of squares
Source: 2021 Iberoamerican Mathematical Olympiad, P4
October 21, 2021
algebra
Problem Statement
Let
a
,
b
,
c
,
x
,
y
,
z
a,b,c,x,y,z
a
,
b
,
c
,
x
,
y
,
z
be real numbers such that
a
2
+
x
2
=
b
2
+
y
2
=
c
2
+
z
2
=
(
a
+
b
)
2
+
(
x
+
y
)
2
=
(
b
+
c
)
2
+
(
y
+
z
)
2
=
(
c
+
a
)
2
+
(
z
+
x
)
2
a^2+x^2=b^2+y^2=c^2+z^2=(a+b)^2+(x+y)^2=(b+c)^2+(y+z)^2=(c+a)^2+(z+x)^2
a
2
+
x
2
=
b
2
+
y
2
=
c
2
+
z
2
=
(
a
+
b
)
2
+
(
x
+
y
)
2
=
(
b
+
c
)
2
+
(
y
+
z
)
2
=
(
c
+
a
)
2
+
(
z
+
x
)
2
Show that
a
2
+
b
2
+
c
2
=
x
2
+
y
2
+
z
2
a^2+b^2+c^2=x^2+y^2+z^2
a
2
+
b
2
+
c
2
=
x
2
+
y
2
+
z
2
.
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