Difference between revisions of "1993 USAMO Problems/Problem 1"
ZzZzZzZzZzZz (talk | contribs) (New page: ==Problem== For each integer <math>n\ge 2</math>, determine, with proof, which of the two positive real numbers <math>a</math> and <math>b</math> satisfying <cmath>a^n=a+1,\qquad b^{2n}=b+...) |
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Revision as of 14:57, 22 March 2008
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 .