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2020 Junior Balkаn MO
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2020 Junior Balkаn MO
Problems
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Junior Ballkan Mathematical Olympiad 2020, Problem 1
Source: Junior Ballkan Mathematical Olympiad 2020, Problem 1
9/11/2020
Find all triples
(
a
,
b
,
c
)
(a,b,c)
(
a
,
b
,
c
)
of real numbers such that the following system holds:
{
a
+
b
+
c
=
1
a
+
1
b
+
1
c
a
2
+
b
2
+
c
2
=
1
a
2
+
1
b
2
+
1
c
2
\begin{cases} a+b+c=\frac{1}{a}+\frac{1}{b}+\frac{1}{c} \\a^2+b^2+c^2=\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}\end{cases}
{
a
+
b
+
c
=
a
1
+
b
1
+
c
1
a
2
+
b
2
+
c
2
=
a
2
1
+
b
2
1
+
c
2
1
Proposed by Dorlir Ahmeti, Albania
algebra
system of equations
Junior Balkan
polynomial
Junior
JBMO
jbmo2020