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Ireland National Math Olympiad
2014 Irish Math Olympiad
4
4
Part of
2014 Irish Math Olympiad
Problems
(1)
Irish Mathematical Olympiad 2014 (1)
Source: Paper 1 , Problem 4
9/6/2014
Three different non-zero real numbers
a
,
b
,
c
a,b,c
a
,
b
,
c
satisfy the equations
a
+
2
b
=
b
+
2
c
=
c
+
2
a
=
p
a+\frac{2}{b}=b+\frac{2}{c}=c+\frac{2}{a}=p
a
+
b
2
=
b
+
c
2
=
c
+
a
2
=
p
, where
p
p
p
is a real number. Prove that
a
b
c
+
2
p
=
0.
abc+2p=0.
ab
c
+
2
p
=
0.
algebra proposed
algebra