2007 AIME II Problems/Problem 14
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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 • Resources) | ||
| Preceded by Problem 13 | Followed by Problem 15 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
