1989 USAMO Problems/Problem 5
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Problem
Let 
and 
be real numbers such that
Determine, with proof, which of the two numbers, or , is larger.
Solution
The answer is .
We define real functions 
and 
as follows:
We wish to show that if 
, then 
.
We first note that when 
, 
, 
, and 
, so
Similarly, 
.
We also note that if 
, then
Similarly 
. It then follows that 
.
Now, for all 
,
Since and are both strictly increasing functions over the nonnegative reals, it then follows that
so 
, as desired. 
Resources
| 1989 USAMO (Problems) | ||
| Preceded by Problem 4 | 1 • 2 • 3 • 4 • 5 | Followed by Final Question |
| All USAMO Problems and Solutions | ||
