1976 AHSME Problems/Problem 27
Contents
Problem
If then equals
Solution 1
Let and
Note that Since we have
On the other hand, note that Since we have
Finally, the answer is
~Someonenumber011 (Fundamental Logic)
~MRENTHUSIASM (Reconstruction)
Solution 2
Let and
Note that We rewrite each term in the numerator separately:
- Let for some nonnegative rational numbers and We square both sides of this equation, then rearrange:
It follows that
By inspection, we get Alternatively, we conclude that and are the solutions to the quadratic equation by Vieta's Formulas, in which
Therefore, we obtain
- Similarly, we obtain
Substituting these results into we have On the other hand, we have by the argument of either Solution 1 or Solution 2.
Finally, the answer is
~MRENTHUSIASM
See also
1976 AHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 26 |
Followed by Problem 28 | |
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