Difference between revisions of "1984 AIME Problems/Problem 15"
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Determine <math>\displaystyle w^2+x^2+y^2+z^2</math> if | Determine <math>\displaystyle w^2+x^2+y^2+z^2</math> if | ||
− | <center><math> \frac{x^2}{2^2-1}+\frac{y^2}{2^2-3^2}+\frac{z^2}{2^2-5^2}+\frac{w^2}{2^2-7^2}=1 </math></ | + | <div style="text-align:center;"><math> \frac{x^2}{2^2-1}+\frac{y^2}{2^2-3^2}+\frac{z^2}{2^2-5^2}+\frac{w^2}{2^2-7^2}=1 </math><br /><math> \frac{x^2}{4^2-1}+\frac{y^2}{4^2-3^2}+\frac{z^2}{4^2-5^2}+\frac{w^2}{4^2-7^2}=1 </math><br /><math> \frac{x^2}{6^2-1}+\frac{y^2}{6^2-3^2}+\frac{z^2}{6^2-5^2}+\frac{w^2}{6^2-7^2}=1 </math><br /><math> \frac{x^2}{8^2-1}+\frac{y^2}{8^2-3^2}+\frac{z^2}{8^2-5^2}+\frac{w^2}{8^2-7^2}=1 </math></div> |
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== Solution == | == Solution == |
Revision as of 21:02, 10 September 2007
Problem
Determine if
![$\frac{x^2}{2^2-1}+\frac{y^2}{2^2-3^2}+\frac{z^2}{2^2-5^2}+\frac{w^2}{2^2-7^2}=1$](http://latex.artofproblemsolving.com/3/f/a/3faf3bd2051d28e84510522f1060d255643f46fe.png)
![$\frac{x^2}{4^2-1}+\frac{y^2}{4^2-3^2}+\frac{z^2}{4^2-5^2}+\frac{w^2}{4^2-7^2}=1$](http://latex.artofproblemsolving.com/9/d/2/9d236639c64e0cd9eeb0bc55dcadef28a7fbb48a.png)
![$\frac{x^2}{6^2-1}+\frac{y^2}{6^2-3^2}+\frac{z^2}{6^2-5^2}+\frac{w^2}{6^2-7^2}=1$](http://latex.artofproblemsolving.com/d/9/0/d907da2778c251c183225c514cdab8b8f6a62e97.png)
![$\frac{x^2}{8^2-1}+\frac{y^2}{8^2-3^2}+\frac{z^2}{8^2-5^2}+\frac{w^2}{8^2-7^2}=1$](http://latex.artofproblemsolving.com/c/3/f/c3fed85bdaf16d0b43eb458a87be68b5d851b31b.png)
Solution
For each of the values , we have the equation
However, each side of the equation is a polynomial in of degree at most 3, and they have 4 common roots. Therefore, the polynomials must be equal.
Now we can plug in into the polynomial equation. Most terms drop, and we end up with
so that
Similarly, we can plug in and get
Now add them up...
with a sum of
See also
1984 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 |