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AIME Problems
1997 AIME Problems
9
9
Part of
1997 AIME Problems
Problems
(1)
Fractional Parts
Source:
12/20/2005
Given a nonnegative real number
x
,
x,
x
,
let
⟨
x
⟩
\langle x\rangle
⟨
x
⟩
denote the fractional part of
x
;
x;
x
;
that is,
⟨
x
⟩
=
x
−
⌊
x
⌋
,
\langle x\rangle=x-\lfloor x\rfloor,
⟨
x
⟩
=
x
−
⌊
x
⌋
,
where
⌊
x
⌋
\lfloor x\rfloor
⌊
x
⌋
denotes the greatest integer less than or equal to
x
.
x.
x
.
Suppose that
a
a
a
is positive,
⟨
a
−
1
⟩
=
⟨
a
2
⟩
,
\langle a^{-1}\rangle=\langle a^2\rangle,
⟨
a
−
1
⟩
=
⟨
a
2
⟩
,
and
2
<
a
2
<
3.
2<a^2<3.
2
<
a
2
<
3.
Find the value of
a
12
−
144
a
−
1
.
a^{12}-144a^{-1}.
a
12
−
144
a
−
1
.
floor function