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Difference between revisions of "2022 SSMO Relay Round 4 Problems"
 
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==Problem 2==
 
==Problem 2==
  
−
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.
  
 
[[2022 SSMO Relay Round 4 Problems/Problem 2|Solution]]
 
[[2022 SSMO Relay Round 4 Problems/Problem 2|Solution]]
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==Problem 3==
 
==Problem 3==
  
−
Let <math>T=</math> TNYWR. If <math>f(1)=1</math>, <math>f(2)=12</math>, and <cmath>f(n+2)=12f(n+1)-20f(n)</cmath> for all positive integers <math>n</math>, find the remainder when <math>f(T)</math> is divided by <math>1000.</math>
+
Let <math>T=TNYWR</math>. If <math>f(1)=1</math>, <math>f(2)=12</math>, and <cmath>f(n+2)=12f(n+1)-20f(n)</cmath> for all positive integers <math>n</math>, find the remainder when <math>f(T)</math> is divided by <math>1000.</math>
  
 
[[2022 SSMO Relay Round 4 Problems/Problem 3|Solution]]
 
[[2022 SSMO Relay Round 4 Problems/Problem 3|Solution]]

Latest revision as of 19:18, 2 May 2025

Problem 1

On any given day, there is a $70\%$ chance that a robot will find a new organism, a $20\%$ chance it will find an already discovered organism, and a $10\%$ chance that it will find nothing. Given that it has found a new organism, there is a $90\%$ chance it will correctly determine that it is a new organism, and given that it has found an already discovered organism, there is a $75\%$ chance that it will correctly determine that it has already been discovered. The expected number of days that the robot will take to report that it has found a new organism (regardless of whether it actually has) can be expressed as $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$

Solution

Problem 2

Let $T=TNYWR.$ The roots of $f(x)=x^3+5x+8$ are $r_1,r_2,r_3.$ Let $g_n(x)$ be a polynomial with roots $r_1+n, r_2+n,r_3+n.$ If \[S=\sum_{n=1}^{T}(-1)^{n}g_n(5),\] find the remainder when $S$ is divided by 1000.

Solution

Problem 3

Let $T=TNYWR$. If $f(1)=1$, $f(2)=12$, and \[f(n+2)=12f(n+1)-20f(n)\] for all positive integers $n$, find the remainder when $f(T)$ is divided by $1000.$

Solution

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