2021 Fall AMC 12A Problems/Problem 25

Problem

Let $m\ge 5$ be an odd integer, and let $D(m)$ denote the number of quadruples $\big(a_1, a_2, a_3, a_4\big)$ of distinct integers with $1\le a_i \le m$ for all $i$ such that $m$ divides $a_1+a_2+a_3+a_4$ . There is a polynomial $q(x) = c_3x^3+c_2x^2+c_1x+c_0$ such that $D(m) = q(m)$ for all odd integers $m\ge 5$ . What is $c_1?$

$(\textbf{A})\ {-}6\qquad(\textbf{B}) \ {-}1\qquad(\textbf{C}) \ 4\qquad(\textbf{D}) \ 6\qquad(\textbf{E}) \ 11$

Solution

WORKING AND WILL FINISH UP SOON. APPRECIATE IT IF NO EDITS.

~MRENTHUSIASM

2021 Fall AMC 12A (Problems • Answer Key • Resources)
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2021 Fall AMC 12A Problems/Problem 25

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