1993 USAMO Problems/Problem 1
Problem
For each integer , determine, with proof, which of the two positive real numbers
and
satisfying
is larger.
Solution
Square and rearrange the first equation and also rearrange the second.
\begin{gather}
a^{2n}-a=a^2+a+1\\
b^{2n}-b=3a
\end{gather}
It is trivial that
\begin{equation}
(a-1)^2>0
\end{equation}
since clearly cannot equal 0 (Otherwise
). Thus
\begin{gather}
a^2+a+1>3a\\
a^{2n}-a>b^{2n}-b
\end{gather}
where we substituted in equations (1) and (2) to achieve (5). If
, then
since
,
, and
are all positive. Adding the two would mean
, a contradiction, so
. However, when
equals 0 or 1, the first equation becomes meaningless, so we conclude that for each integer
, we always have
.