2000 AIME II Problems/Problem 13
Contents
Problem
The equation has exactly two real roots, one of which is , where , and are integers, and are relatively prime, and . Find .
Solution
We may factor the equation as:^{[1]}
Now for real . Thus the real roots must be the roots of the equation . By the quadratic formula the roots of this are:
Thus , and so the final answer is .
^ A well-known technique for dealing with symmetric (or in this case, nearly symmetric) polynomials is to divide through by a power of with half of the polynomial's degree (in this case, divide through by ), and then to use one of the substitutions . In this case, the substitution gives and , which reduces the polynomial to just . Then one can backwards solve for .
Note: After dividing the equation with , divide again with before substituting it with to get it right.
A slightly different approach using symmetry:
Let .
Notice that the equation can be rewritten (after dividing across by ) as
Now it is easy to see that the equation reduces to
so for real solutions we have . Solve the quadratic in to get the final answer as .
Solution 2 (Complex Bash)
It would be really nice if the coefficients were symmetrical. What if we make the substitution, . The the polynomial becomes
It's symmetric! Dividing by and rearranging, we get
Now, if we let , we can get the equations
and
(These come from squaring and subtracting , then multiplying that result by and subtracting ) Plugging this into our polynomial, expanding, and rearranging, we get
Now, we see that the two terms must cancel in order to get this polynomial equal to , so what squared equals 3? Plugging in into the polynomial, we see that it works! Is there something else that equals 3 when squared? Trying , we see that it also works! Great, we use long division on the polynomial by
and we get
.
We know that the other two solutions for z wouldn't result in real solutions for since we have to solve a quadratic with a negative discriminant, then multiply by . We get that . Solving for (using ) we get that , and multiplying this by (because ) we get that for a final answer of
-Grizzy
Solution 3 (Geometric Series)
Observe that the given equation may be rearranged as . The expression in parentheses is a geometric series with common factor . Using the geometric sum formula, we rewrite as . Factoring a bit, we get . Note that setting gives , which is clearly extraneous. Hence, we set and use the quadratic formula to get the desired root
~keeper1098
Solution 4
If we look at the polynomial's terms, we can see that the number of zeros in a term more or less correlates to the power of . Thus, we let . The equation then becomes , or .
By the difference of cubes formula, , so we have two cases: either , or . We start with the second formula as it is simpler.
Solving with the quadratic formula after re-substitution, we see that , so the answer is .
For the sake of completeness, if we check the other equation, we come to the conclusion that , so no real solution exists for . Thus our solution is correct.
~eevee9406
Video solution
https://www.youtube.com/watch?v=mAXDdKX52TM
See also
2000 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 12 |
Followed by Problem 14 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
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