2021 Fall AMC 12A Problems/Problem 25
Problem
Let be an odd integer, and let denote the number of quadruples of distinct integers with for all such that divides . There is a polynomial such that for all odd integers . What is
Solution
For a fixed value of there is a total of possible ordered quadruples
Let We claim that exactly of these ordered quadruples satisfy that divides
Since we conclude that is the complete system of residues modulo for all integers
Given any ordered quadruple in modulo it follows that exactly one of these ordered quadruples satisfy that divides We conclude that so By Vieta's Formulas, we get
~MRENTHUSIASM
See Also
2021 Fall AMC 12A (Problems • Answer Key • Resources) | |
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