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Difference between revisions of "2021 Fall AMC 12A Problems/Problem 25"

(Created page with "==Problem== Let <math>m\ge 5</math> be an odd integer, and let <math>D(m)</math> denote the number of quadruples <math>\big(a_1, a_2, a_3, a_4\big)</math> of distinct integers...")
 
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==Problem==
 
==Problem==
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Let <math>m\ge 5</math> be an odd integer, and let <math>D(m)</math> denote the number of quadruples <math>\big(a_1, a_2, a_3, a_4\big)</math> of distinct integers with <math>1\le a_i \le m</math> for all <math>i</math> such that <math>m</math> divides <math>a_1+a_2+a_3+a_4</math>. There is a polynomial
+
Let <math>m\ge 5</math> be an odd integer, and let <math>D(m)</math> denote the number of quadruples <math>(a_1, a_2, a_3, a_4)</math> of distinct integers with <math>1\le a_i \le m</math> for all <math>i</math> such that <math>m</math> divides <math>a_1+a_2+a_3+a_4</math>. There is a polynomial
 
<cmath>q(x) = c_3x^3+c_2x^2+c_1x+c_0</cmath>such that <math>D(m) = q(m)</math> for all odd integers <math>m\ge 5</math>. What is <math>c_1?</math>
 
<cmath>q(x) = c_3x^3+c_2x^2+c_1x+c_0</cmath>such that <math>D(m) = q(m)</math> for all odd integers <math>m\ge 5</math>. What is <math>c_1?</math>
  
−
<math>(\textbf{A})\ {-}6\qquad(\textbf{B}) \ {-}1\qquad(\textbf{C}) \ 4\qquad(\textbf{D}) \ 6\qquad(\textbf{E}) \ 11</math>
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<math>\textbf{(A)}\ {-}6\qquad\textbf{(B)}\ {-}1\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 6\qquad\textbf{(E)}\ 11</math>
  
 
==Solution==
 
==Solution==
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<b>WORKING AND WILL FINISH UP SOON. APPRECIATE IT IF NO EDITS.</b>
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For a fixed value of <math>m,</math> there is a total of <math>m(m-1)(m-2)(m-3)</math> possible ordered quadruples <math>(a_1, a_2, a_3, a_4).</math>
 +
 
 +
Let <math>S=a_1+a_2+a_3+a_4.</math> We claim that exactly <math>\frac1m</math> of these ordered quadruples satisfy that <math>m</math> divides <math>S:</math>
 +
 
 +
Since <math>\gcd(m,4)=1,</math> we conclude that <cmath>\{k+4(0),k+4(1),k+4(2),\ldots,k+4(m-1)\}</cmath> is the complete system of residues modulo <math>m</math> for all integers <math>k.</math>
 +
 
 +
Given any ordered quadruple <math>(a'_1, a'_2, a'_3, a'_4)</math> in modulo <math>m,</math> it follows that exactly one of these <math>m</math> ordered quadruples satisfy that <math>m</math> divides <math>S:</math>
 +
<cmath>\begin{array}{c|c}
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& \\ [-2.5ex]
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\textbf{Ordered Quadruple} & \textbf{Sum Modulo }\boldsymbol{m} \\ [0.5ex]
 +
\hline
 +
& \\ [-2ex]
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(a'_1, a'_2, a'_3, a'_4) & S+4(0) \\ [0.5ex]
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(a'_1+1, a'_2+1, a'_3+1, a'_4+1) & S+4(1) \\ [0.5ex]
 +
(a'_1+2, a'_2+2, a'_3+2, a'_4+2) & S+4(2) \\ [0.5ex]
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\cdots & \cdots \\ [0.5ex]
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(a'_1+m-1, a'_2+m-1, a'_3+m-1, a'_4+m-1) & S+4(m-1) \\ [0.5ex]
 +
\end{array}</cmath>
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We conclude that <math>q(m)=\frac1m\cdot[m(m-1)(m-2)(m-3)]=(m-1)(m-2)(m-3),</math> so <cmath>q(x)=(x-1)(x-2)(x-3)=c_3x^3+c_2x^2+c_1x+c_0.</cmath>
 +
By Vieta's Formulas, we get <math>c_1=1\cdot2+1\cdot3+2\cdot3=\boxed{\textbf{(E)}\ 11}.</math>
  
 
~MRENTHUSIASM
 
~MRENTHUSIASM

Revision as of 01:18, 24 November 2021

Problem

Let $m\ge 5$ be an odd integer, and let $D(m)$ denote the number of quadruples $(a_1, a_2, a_3, a_4)$ 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

For a fixed value of $m,$ there is a total of $m(m-1)(m-2)(m-3)$ possible ordered quadruples $(a_1, a_2, a_3, a_4).$

Let $S=a_1+a_2+a_3+a_4.$ We claim that exactly $\frac1m$ of these ordered quadruples satisfy that $m$ divides $S:$

Since $\gcd(m,4)=1,$ we conclude that \[\{k+4(0),k+4(1),k+4(2),\ldots,k+4(m-1)\}\] is the complete system of residues modulo $m$ for all integers $k.$

Given any ordered quadruple $(a'_1, a'_2, a'_3, a'_4)$ in modulo $m,$ it follows that exactly one of these $m$ ordered quadruples satisfy that $m$ divides $S:$ \[\begin{array}{c|c} & \\ [-2.5ex] \textbf{Ordered Quadruple} & \textbf{Sum Modulo }\boldsymbol{m} \\ [0.5ex] \hline & \\ [-2ex] (a'_1, a'_2, a'_3, a'_4) & S+4(0) \\ [0.5ex] (a'_1+1, a'_2+1, a'_3+1, a'_4+1) & S+4(1) \\ [0.5ex] (a'_1+2, a'_2+2, a'_3+2, a'_4+2) & S+4(2) \\ [0.5ex] \cdots & \cdots \\ [0.5ex] (a'_1+m-1, a'_2+m-1, a'_3+m-1, a'_4+m-1) & S+4(m-1) \\ [0.5ex] \end{array}\] We conclude that $q(m)=\frac1m\cdot[m(m-1)(m-2)(m-3)]=(m-1)(m-2)(m-3),$ so \[q(x)=(x-1)(x-2)(x-3)=c_3x^3+c_2x^2+c_1x+c_0.\] By Vieta's Formulas, we get $c_1=1\cdot2+1\cdot3+2\cdot3=\boxed{\textbf{(E)}\ 11}.$

~MRENTHUSIASM

See Also

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