1993 AIME Problems/Problem 5
Contents
Problem
Let . For integers , define . What is the coefficient of in ?
Solution 1
Notice that
Using the formula for the sum of the first numbers, . Therefore,
Substituting into the function definition, we get . We only need the coefficients of the linear terms, which we can find by the binomial theorem.
- will have a linear term of .
- will have a linear term of .
- will have a linear term of .
Adding up the coefficients, we get .
Solution 2
Notice the transformation of adds to the roots. Thus, all these transformations will take the roots and add to them. (Indeed, this is very easy to check in general.)
Let the roots be Then By Vieta's/expanding/common sense, you see the coefficient of is Expanding yields Using Vieta's (again) and plugging stuff in yields
See also
1993 AIME (Problems • Answer Key • Resources) | ||
Preceded by Problem 4 |
Followed by Problem 6 | |
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