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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