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.
It is trivial that
since
clearly cannot equal
(Otherwise
). Thus
where we substituted in equations (1) and (2) to achieve (5). Notice that from
we have
. Thus, if
, then
a>1\Rightarrow a^{2n-1}-1>0
b^{2n}-b>a^{2n}-a
a> b
n
0
1
n\ge 2
a>b$.
See also
1993 USAMO (Problems • Resources) | ||
Preceded by First question |
Followed by Problem 2 | |
1 • 2 • 3 • 4 • 5 | ||
All USAMO Problems and Solutions |