2000 AMC 12 Problems/Problem 15: Difference between revisions
No edit summary |
duplicate |
||
| Line 1: | Line 1: | ||
{{duplicate|[[2000 AMC 12 Problems|2000 AMC 12 #15]] and [[2000 AMC 10 Problems|2000 AMC 10 #24]]}} | |||
== Problem == | == Problem == | ||
Let <math>f</math> be a [[function]] for which <math>f(x/3) = x^2 + x + 1</math>. Find the sum of all values of <math>z</math> for which <math>f(3z) = 7</math>. | Let <math>f</math> be a [[function]] for which <math>f(x/3) = x^2 + x + 1</math>. Find the sum of all values of <math>z</math> for which <math>f(3z) = 7</math>. | ||
| Line 11: | Line 13: | ||
== See also == | == See also == | ||
{{AMC12 box|year=2000|num-b=14|num-a=16}} | {{AMC12 box|year=2000|num-b=14|num-a=16}} | ||
{{AMC10 box|year=2000|num-b=23|num-a=25}} | |||
[[Category:Introductory Algebra Problems]] | [[Category:Introductory Algebra Problems]] | ||
Revision as of 23:07, 26 November 2011
- The following problem is from both the 2000 AMC 12 #15 and 2000 AMC 10 #24, so both problems redirect to this page.
Problem
Let 
be a function for which 
. Find the sum of all values of 
for which 
.
Solution
Let 
; then 
. Thus 
, and 
. These sum up to 
.
Alternative solution: When we have 
, we just use Vieta's and get the sum is 
See also
| 2000 AMC 12 (Problems • Answer Key • Resources) | |
| Preceded by Problem 14 |
Followed by Problem 16 |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
| All AMC 12 Problems and Solutions | |
| 2000 AMC 10 (Problems • Answer Key • Resources) | ||
| Preceded by Problem 23 |
Followed by Problem 25 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
| All AMC 10 Problems and Solutions | ||
