2007 AIME II Problems/Problem 14
Problem
Let be a polynomial with real coefficients such that
and for all
,
Find
Solution
- Note:The following solution(s) are non-rigorous.
Substitute the values . We find that
, and that
. This means that
. This suggests that
are roots of the polynomial, and so
will be a root of the polynomial.
The polynomial is likely in the form of ;
appears to satisfy the same relation as
, so it also probably has the same roots. Thus,
is the solution. Guessing values for
, try
. Checking a couple of values shows that
works, and so the solution is
.
See also
2007 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 13 |
Followed by Problem 15 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |