Difference between revisions of "2022 SSMO Relay Round 4 Problems/Problem 2"
(Created page with "==Problem== The roots of <math>f(x)=x^3+5x+8</math> are <math>r_1,r_2,r_3.</math> Let <math>g_n(x)</math> be a polynomial with roots <math>r_1+n, r_2+n,r_3+n.</math> If<cmath>...") |
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==Problem== | ==Problem== | ||
| − | The roots of <math>f(x)=x^3+5x+8</math> are <math>r_1,r_2,r_3.</math> Let <math>g_n(x)</math> be a polynomial with roots <math>r_1+n, r_2+n,r_3+n.</math> If<cmath>S=\sum_{n=1}^{T}(-1)^{n}g_n(5),</cmath>find the remainder when <math>S</math> is divided by 1000. | + | Let <math>T=TNYWR</math>. The roots of <math>f(x)=x^3+5x+8</math> are <math>r_1,r_2,r_3.</math> Let <math>g_n(x)</math> be a polynomial with roots <math>r_1+n, r_2+n,r_3+n.</math> If<cmath>S=\sum_{n=1}^{T}(-1)^{n}g_n(5),</cmath>find the remainder when <math>S</math> is divided by 1000. |
==Solution== | ==Solution== | ||
Latest revision as of 19:22, 2 May 2025
Problem
Let 
. The roots of 
are 
Let 
be a polynomial with roots 
If![\[S=\sum_{n=1}^{T}(-1)^{n}g_n(5),\]](http://latex.artofproblemsolving.com/d/4/5/d4599d33d3a9b9d0c86103b46aa54cff389988a7.png)
find the remainder when 
is divided by 1000.
