MathDB
Problems
Contests
National and Regional Contests
Korea Contests
Korea Junior Mathematics Olympiad
2011 Korea Junior Math Olympiad
1
A three variable equations
A three variable equations
Source: KJMO 2011 pr 1
October 17, 2017
algebra
Problem Statement
Real numbers
a
a
a
,
b
b
b
,
c
c
c
which are differ from
1
1
1
satisfies the following conditions; (1)
a
b
c
=
1
abc =1
ab
c
=
1
(2)
a
2
+
b
2
+
c
2
−
(
1
a
2
+
1
b
2
+
1
c
2
)
=
8
(
a
+
b
+
c
)
−
8
(
a
b
+
b
c
+
c
a
)
a^2+b^2+c^2 - \left( \dfrac{1}{a^2} + \dfrac{1}{b^2} + \dfrac{1}{c^2} \right) = 8(a+b+c) - 8 (ab+bc+ca)
a
2
+
b
2
+
c
2
−
(
a
2
1
+
b
2
1
+
c
2
1
)
=
8
(
a
+
b
+
c
)
−
8
(
ab
+
b
c
+
c
a
)
Find all possible values of expression
1
a
−
1
+
1
b
−
1
+
1
c
−
1
\dfrac{1}{a-1} + \dfrac{1}{b-1} + \dfrac{1}{c-1}
a
−
1
1
+
b
−
1
1
+
c
−
1
1
.
Back to Problems
View on AoPS