Difference between revisions of "1997 JBMO Problems/Problem 2"
Rockmanex3 (talk | contribs) (Solution to Problem 2) |
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== Problem == | == Problem == | ||
− | + | Let <math>\frac{x^2+y^2}{x^2-y^2} + \frac{x^2-y^2}{x^2+y^2} = k</math>. Compute the following expression in terms of <math>k</math>: | |
<cmath> E(x,y) = \frac{x^8 + y^8}{x^8-y^8} - \frac{ x^8-y^8}{x^8+y^8}. </cmath> | <cmath> E(x,y) = \frac{x^8 + y^8}{x^8-y^8} - \frac{ x^8-y^8}{x^8+y^8}. </cmath> | ||
== Solution == | == Solution == | ||
− | == See | + | To start, we add the two fractions and simplify. |
+ | <cmath>\begin{align*} | ||
+ | k &= \frac{(x^2+y^2)^2 + (x^2-y^2)^2}{x^4-y^4} \\ | ||
+ | &= \frac{2x^4 + 2y^4}{x^4 - y^4}. | ||
+ | \end{align*}</cmath> | ||
+ | Dividing both sides by two yields | ||
+ | <cmath>\frac{k}{2} = \frac{x^4 + y^4}{x^4 - y^4}.</cmath> | ||
+ | That means | ||
+ | <cmath>\begin{align*} | ||
+ | \frac{x^4 + y^4}{x^4 - y^4} + \frac{x^4 - y^4}{x^4 + y^4} &= \frac{k}{2} + \frac{2}{k} \\ | ||
+ | \frac{(x^4 + y^4)^2 + (x^4 - y^4)^2}{x^8 - y^8} &= \frac{k^2 + 4}{2k} \\ | ||
+ | \frac{2x^8 + 2y^8}{x^8 - y^8} &= \frac{k^2 + 4}{2k}. | ||
+ | \end{align*}</cmath> | ||
+ | Dividing both sides by two yields | ||
+ | <cmath>\frac{x^8 + y^8}{x^8 - y^8} = \frac{k^2 + 4}{4k}.</cmath> | ||
+ | That means | ||
+ | <cmath>\begin{align*} | ||
+ | \frac{x^8 + y^8}{x^8 - y^8} - \frac{x^8 - y^8}{x^8 + y^8} &= \frac{k^2 + 4}{4k} - \frac{4k}{k^2 + 4} \\ | ||
+ | &= \frac{k^4 + 8k^2 + 16 - 16k^2}{4k(k^2 + 4)} \\ | ||
+ | &= \frac{k^4 - 8k^2 + 16}{4k(k^2 + 4)} \\ | ||
+ | &= \boxed{\frac{(k^2 - 4)^2}{4k(k^2 + 4)}}. | ||
+ | \end{align*}</cmath> | ||
+ | |||
+ | == See Also == | ||
{{JBMO box|year=1997|num-b=1|num-a=3}} | {{JBMO box|year=1997|num-b=1|num-a=3}} | ||
[[Category:Intermediate Algebra Problems]] | [[Category:Intermediate Algebra Problems]] |
Latest revision as of 13:49, 4 August 2018
Problem
Let . Compute the following expression in terms of :
Solution
To start, we add the two fractions and simplify. Dividing both sides by two yields That means Dividing both sides by two yields That means
See Also
1997 JBMO (Problems • Resources) | ||
Preceded by Problem 1 |
Followed by Problem 3 | |
1 • 2 • 3 • 4 • 5 | ||
All JBMO Problems and Solutions |