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National and Regional Contests
Ireland Contests
Ireland National Math Olympiad
1989 Irish Math Olympiad
4
4
Part of
1989 Irish Math Olympiad
Problems
(1)
positive real number (IrMO 1989)
Source:
1/17/2014
Let
a
a
a
be a positive real number and let
b
=
a
+
a
2
+
1
3
+
a
−
a
2
+
1
3
b= \sqrt[3] {a+ \sqrt {a^{2}+1}} + \sqrt[3] {a- \sqrt {a^{2}+1}}
b
=
3
a
+
a
2
+
1
+
3
a
−
a
2
+
1
.Prove that
b
b
b
is a positive integer if, and only if,
a
a
a
is a positive integer of the form
1
2
n
(
n
2
+
3
)
\frac{1}{2} n(n^{2}+3)
2
1
n
(
n
2
+
3
)
, for some positive integer
n
n
n
.
algebra unsolved
algebra