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All-Russian Olympiad Regional Round
1997 All-Russian Olympiad Regional Round
10.8
\sqrt{x + a} +\sqrt{y+b}+\sqrt{z + c} - All-Russian MO 1997 Regional (R4) 10.8
\sqrt{x + a} +\sqrt{y+b}+\sqrt{z + c} - All-Russian MO 1997 Regional (R4) 10.8
Source:
September 24, 2024
algebra
radical
Problem Statement
Prove that if
x
+
a
+
y
+
b
+
z
+
c
=
y
+
a
+
z
+
b
+
x
+
c
=
z
+
a
+
x
+
b
+
y
+
c
\sqrt{x + a} +\sqrt{y+b}+\sqrt{z + c} =\sqrt{y + a} +\sqrt{z + b} +\sqrt{x + c} =\sqrt{z + a} +\sqrt{x+b}+\sqrt{y+c}
x
+
a
+
y
+
b
+
z
+
c
=
y
+
a
+
z
+
b
+
x
+
c
=
z
+
a
+
x
+
b
+
y
+
c
for some
a
,
b
,
c
,
x
,
y
,
z
a, b, c, x, y, z
a
,
b
,
c
,
x
,
y
,
z
, then
x
=
y
=
z
x = y = z
x
=
y
=
z
or
a
=
b
=
c
a = b = c
a
=
b
=
c
.
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