Difference between revisions of "1990 AIME Problems/Problem 15"
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== Problem == | == Problem == | ||
Find <math>a_{}^{}x^5 + b_{}y^5</math> if the real numbers <math>a_{}^{}</math>, <math>b_{}^{}</math>, <math>x_{}^{}</math>, and <math>y_{}^{}</math> satisfy the equations | Find <math>a_{}^{}x^5 + b_{}y^5</math> if the real numbers <math>a_{}^{}</math>, <math>b_{}^{}</math>, <math>x_{}^{}</math>, and <math>y_{}^{}</math> satisfy the equations | ||
| − | < | + | <cmath>ax + by = 3^{}_{},</cmath> |
| − | < | + | <cmath>ax^2 + by^2 = 7^{}_{},</cmath> |
| − | < | + | <cmath>ax^3 + by^3 = 16^{}_{},</cmath> |
| − | < | + | <cmath>ax^4 + by^4 = 42^{}_{}.</cmath> |
== Solution == | == Solution == | ||
Revision as of 18:28, 15 October 2007
Problem
Find 
if the real numbers 
, 
, 
, and 
satisfy the equations
![\[ax + by = 3^{}_{},\]](http://latex.artofproblemsolving.com/c/d/5/cd552c7ada95d98bc5f7df21d63d11b3c4fbaa64.png)
![\[ax^2 + by^2 = 7^{}_{},\]](http://latex.artofproblemsolving.com/4/a/a/4aa8b398fa738219f22d7c40be8ac8888969079a.png)
![\[ax^3 + by^3 = 16^{}_{},\]](http://latex.artofproblemsolving.com/a/e/9/ae92af789a106a02e74702b5b7bb0fce7fbf3525.png)
![\[ax^4 + by^4 = 42^{}_{}.\]](http://latex.artofproblemsolving.com/a/c/6/ac6ea536de1ad0908972237419f7531073e2cbca.png)
Solution
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See also
| 1990 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 | ||
