2023 AIME I Problems/Problem 9
Find the number of cubic polynomials , where , , and are integers in ${−20, −10, −18, . . . , 18, 19, 20}$ (Error compiling LaTeX. Unknown error_msg), such that there is a unique integer with .
Find the number of cubic polynomials , where , , and are integers in , such that there is a unique integer with .
Find the number of cubic polynomials , where , , and are integers in , such that there is a unique integer with .
Solution
It can be easily noticed that is independent of the condition , and can thus safely take all possible values between and .
There are two possible ways for to be the only integer satisfying : has a double root at or a double root at .
Case 1: has a double root at :
In this case, , or . Thus ranges from to . One of these values, corresponds to a triple root at , which means . Thus there are possible values of . (It can be verified that is an integer).
Case 2: has a double root at :
See the above solution. This yields possible combinations of and .
Thus, in total we have combinations of .
-Alex_Z