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Ireland National Math Olympiad
1988 Irish Math Olympiad
7
7
Part of
1988 Irish Math Olympiad
Problems
(1)
Cubic with Horizontal Chord
Source: 1988 IrMO Paper 1 Problem 7
9/28/2017
A function
f
f
f
, defined on the set of real numbers
R
\mathbb{R}
R
is said to have a horizontal chord of length
a
>
0
a>0
a
>
0
if there is a real number
x
x
x
such that
f
(
a
+
x
)
=
f
(
x
)
f(a+x)=f(x)
f
(
a
+
x
)
=
f
(
x
)
. Show that the cubic f(x)=x^3-x (x\in \mathbb{R}) has a horizontal chord of length
a
a
a
if, and only if,
0
<
a
≤
2
0<a\le 2
0
<
a
≤
2
.
algebra