Difference between revisions of "2022 SSMO Speed Round Problems/Problem 7"

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==Problem==
 
==Problem==
At FenZhu High School, <math>7</math>th graders have a 60\% of chance of having a dog and <math>8</math>th graders have a 40\% chance of having a dog. Suppose there is a classroom of <math>30</math> <math>7</math>th grader and <math>10</math> <math>8</math>th graders. If exactly one person owns a dog, then the probability that a <math>7</math>th grader owns the dog is <math>\frac{m}{n},</math> for relatively prime positive integers <math>m</math> and <math>n.</math> Find <math>m+n.</math>
+
 
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Let <math>A_1=(1, 0)</math>. Define <math>A_{n+1}</math> as the image of <math>A_n</math> under a rotation of either <math>45^{\circ}</math>, <math>90^{\circ}</math>, or <math>135^{\circ}</math> clockwise about the origin, with each choice having a <math>\frac{1}{3}</math> chance of being selected. Find the expected value of the smallest positive integer <math>n>1</math> such that <math>A_n</math> coincides with <math>A_1</math>.
  
 
==Solution==
 
==Solution==
The probability that a <math>7</math>th grader has the only dog is
 
<cmath>
 
    A = \left(30 \cdot \left(\frac{2}{5}\right)^{29} \cdot \frac{3}{5}\right) \cdot \left(\frac{3}{5}\right)^{10}
 
</cmath>
 
and the probability for an <math>8</math>th grader is
 
<cmath>
 
    B = \left(\frac{2}{5}\right)^{30} \cdot \left(10 \cdot \left(\frac{3}{5}\right)^{9} \cdot \frac{2}{5}\right).
 
</cmath>
 
 
Then,
 
<cmath>
 
    \frac{A}{B} = \frac{30 \cdot 2^{29} \cdot 3 \cdot 3^{10}}{2^{30} \cdot 10 \cdot 3^9 \cdot 2} = \frac{30 \cdot 3 \cdot 3}{2 \cdot 10 \cdot 2} = \frac{27}{4}.
 
</cmath>
 
 
The probability is thus <math>\frac{A}{A + B} = \frac{27}{31}</math> so the
 
sum is <math>27 + 31 = \boxed{58}</math>
 

Latest revision as of 19:14, 2 May 2025

Problem

Let $A_1=(1, 0)$. Define $A_{n+1}$ as the image of $A_n$ under a rotation of either $45^{\circ}$, $90^{\circ}$, or $135^{\circ}$ clockwise about the origin, with each choice having a $\frac{1}{3}$ chance of being selected. Find the expected value of the smallest positive integer $n>1$ such that $A_n$ coincides with $A_1$.

Solution