Difference between revisions of "2002 AMC 12P Problems/Problem 21"
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Rewriting the left-hand side gives <cmath>\frac {2(x+y)}{xy} = \frac {9}{x+y}</cmath> | Rewriting the left-hand side gives <cmath>\frac {2(x+y)}{xy} = \frac {9}{x+y}</cmath> | ||
+ | |||
+ | Cross-multiplying gives <math>2(x+y)^2 = 9xy</math> or <cmath>2x^2 - 5xy + 2y^2 = 0</cmath> | ||
== See also == | == See also == | ||
{{AMC12 box|year=2002|ab=P|num-b=20|num-a=22}} | {{AMC12 box|year=2002|ab=P|num-b=20|num-a=22}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Revision as of 14:58, 10 March 2024
Problem
Let and be real numbers greater than for which there exists a positive real number different from , such that
Find the largest possible value of
Solution
We may rewrite the given equation as Since , we have , so we may divide by on both sides. After making the substitutions and , our equation becomes
Rewriting the left-hand side gives
Cross-multiplying gives or
See also
2002 AMC 12P (Problems • Answer Key • Resources) | |
Preceded by Problem 20 |
Followed by Problem 22 |
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 |
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