Difference between revisions of "2000 AMC 12 Problems/Problem 11"
m (→Solution) |
Piemax2713 (talk | contribs) m (→Problem) |
||
Line 1: | Line 1: | ||
{{duplicate|[[2000 AMC 12 Problems|2000 AMC 12 #11]] and [[2000 AMC 10 Problems|2000 AMC 10 #15]]}} | {{duplicate|[[2000 AMC 12 Problems|2000 AMC 12 #11]] and [[2000 AMC 10 Problems|2000 AMC 10 #15]]}} | ||
− | ==Problem== | + | ==Problem=== |
Two non-zero [[real number]]s, <math>a</math> and <math>b,</math> satisfy <math>ab = a - b</math>. Which of the following is a possible value of <math>\frac {a}{b} + \frac {b}{a} - ab</math>? | Two non-zero [[real number]]s, <math>a</math> and <math>b,</math> satisfy <math>ab = a - b</math>. Which of the following is a possible value of <math>\frac {a}{b} + \frac {b}{a} - ab</math>? | ||
Revision as of 19:44, 13 November 2019
- The following problem is from both the 2000 AMC 12 #11 and 2000 AMC 10 #15, so both problems redirect to this page.
Contents
[hide]Problem=
Two non-zero real numbers, and satisfy . Which of the following is a possible value of ?
Solution
.
Another way is to solve the equation for giving then substituting this into the expression and simplifying gives the answer of
Solution 2
This simplifies to . The two integer solutions to this are and . The problem states than and are non-zero, so we consider the case of . So, we end up with
See also
2000 AMC 12 (Problems • Answer Key • Resources) | |
Preceded by Problem 10 |
Followed by Problem 12 |
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 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 10 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.