MathDB
Polynomial Problem

Source: 2021 AMC12A #25

November 11, 2021
algebrapolynomialAMCAMC 12AMC 12 A

Problem Statement

Let m5m\ge 5 be an odd integer, and let D(m)D(m) denote the number of quadruples (a1,a2,a3,a4)\big(a_1, a_2, a_3, a_4\big) of distinct integers with 1aim1\le a_i \le m for all ii such that mm divides a1+a2+a3+a4a_1+a_2+a_3+a_4. There is a polynomial q(x)=c3x3+c2x2+c1x+c0q(x) = c_3x^3+c_2x^2+c_1x+c_0such that D(m)=q(m)D(m) = q(m) for all odd integers m5m\ge 5. What is c1?c_1?
(<spanclass=latexbold>A</span>)6(<spanclass=latexbold>B</span>)1(<spanclass=latexbold>C</span>)4(<spanclass=latexbold>D</span>)6(<spanclass=latexbold>E</span>)11(<span class='latex-bold'>A</span>)\: {-}6\qquad(<span class='latex-bold'>B</span>) \: {-}1\qquad(<span class='latex-bold'>C</span>) \: 4\qquad(<span class='latex-bold'>D</span>) \: 6\qquad(<span class='latex-bold'>E</span>) \: 11