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 | + | Find <math>a_{}^{}x^5 + b_{}y^5</math> if the [[real number]]s <math>a_{}^{}</math>, <math>b_{}^{}</math>, <math>x_{}^{}</math>, and <math>y_{}^{}</math> satisfy the [[equation]]s |
<cmath>ax + by = 3^{}_{},</cmath> | <cmath>ax + by = 3^{}_{},</cmath> | ||
<cmath>ax^2 + by^2 = 7^{}_{},</cmath> | <cmath>ax^2 + by^2 = 7^{}_{},</cmath> | ||
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== See also == | == See also == | ||
{{AIME box|year=1990|num-b=14|after=Last question}} | {{AIME box|year=1990|num-b=14|after=Last question}} | ||
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+ | [[Category:Intermediate Algebra Problems]] |
Revision as of 11:03, 25 November 2007
Problem
Find if the real numbers , , , and satisfy the equations
Solution
Set and . Then the relationship
can be exploited:
Therefore:
Consequently, and . Finally:
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 |