1983 AIME Problems/Problem 9
From AoPSWiki
Problem
Find the minimum value of 
for 
.
Solution
Let 
. We can rewrite the expression as 
.
Since 
and 
because 
, we have 
. So we can apply AM-GM:
The equality holds when 
.
Therefore, the minimum value is 
(when 
; since 
is continuous and increasing on the interval 
and its range on that interval is from 
, by the Intermediate Value Theorem this value is attainable).
See also
| 1983 AIME (Problems • Resources) | ||
| Preceded by Problem 8 | Followed by Problem 10 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
