Difference between revisions of "1976 AHSME Problems/Problem 27"
MRENTHUSIASM (talk | contribs) m |
MRENTHUSIASM (talk | contribs) |
||
Line 11: | Line 11: | ||
Let <math>x=\frac{\sqrt{\sqrt{5}+2}+\sqrt{\sqrt{5}-2}}{\sqrt{\sqrt{5}+1}}</math> and <math>y=\sqrt{3-2\sqrt{2}}.</math> Note that | Let <math>x=\frac{\sqrt{\sqrt{5}+2}+\sqrt{\sqrt{5}-2}}{\sqrt{\sqrt{5}+1}}</math> and <math>y=\sqrt{3-2\sqrt{2}}.</math> Note that | ||
<cmath>\begin{align*} | <cmath>\begin{align*} | ||
− | x^2&=\frac{\sqrt{\sqrt{5}+2}+\sqrt{\sqrt{5}-2}}{\sqrt{\sqrt{5}+1}} | + | x^2&=\frac{\left(\sqrt{\sqrt{5}+2}+\sqrt{\sqrt{5}-2}\right)^2}{\left(\sqrt{\sqrt{5}+1}\right)^2} \\ |
+ | &=\frac{\left(\sqrt{5}+2\right)+2\cdot\left(\sqrt{\sqrt{5}+2}\cdot\sqrt{\sqrt{5}-2}\right)+\left(\sqrt{5}-2\right)}{\sqrt{5}+1} \\ | ||
+ | &=\frac{2\sqrt{5}+2}{\sqrt{5}+1} \\ | ||
+ | &=2. | ||
\end{align*}</cmath> | \end{align*}</cmath> | ||
Revision as of 01:10, 8 September 2021
Problem
If then equals
Solution
Let and Note that
~Someonenumber011 (Solution)
~MRENTHUSIASM (Revision)
See also
1976 AHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 26 |
Followed by Problem 28 | |
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 • 26 • 27 • 28 • 29 • 30 | ||
All AHSME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.