Difference between revisions of "1991 AIME Problems/Problem 15"
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== Solution == | == Solution == | ||
| − | {{ | + | We start by recalling the following simple inequality: Let <math>a_{}^{}</math> and <math>b_{}^{}</math> denote two positive real numbers, then <math>\sqrt{a_{}^{2}+b_{}^{2}}\geq (a+b)/\sqrt{2}</math>, with equality if and only if <math>a_{}^{}=b_{}^{}</math>. |
== See also == | == See also == | ||
{{AIME box|year=1991|num-b=14|after=Last question}} | {{AIME box|year=1991|num-b=14|after=Last question}} | ||
Revision as of 17:37, 19 April 2007
Problem
For positive integer 
, define 
to be the minimum value of the sum
where 
are positive real numbers whose sum is 17. There is a unique positive integer 
for which is also an integer. Find this .
Solution
We start by recalling the following simple inequality: Let 
and 
denote two positive real numbers, then 
, with equality if and only if 
.
See also
| 1991 AIME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 14 |
Followed by Last question | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
