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Difference between revisions of "2022 SSMO Speed Round Problems/Problem 3"

(Created page with "==Problem== Pigs like to eat carrots. Suppose a pig randomly chooses 6 letters from the set <math>\{c,a,r,o,t\}.</math> Then, the pig randomly arranges the 6 letters to form a...")
 
 
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==Problem==
 
==Problem==
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Pigs like to eat carrots. Suppose a pig randomly chooses 6 letters from the set <math>\{c,a,r,o,t\}.</math> Then, the pig randomly arranges the 6 letters to form a "word". If the 6 letters don't spell carrot, the pig gets frustrated and tries to spell it again (by rechoosing the 6 letters and respelling them). What is the expected number of tries it takes for the pig to spell "carrot"?
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Let <math>ABCD</math> be a parallelogram such that <math>E</math> is a point on <math>CD</math> such that <math>\frac{CE}{DE}=\frac{2}{3}.</math> Suppose that <math>BE</math> and <math>AC</math> intersect at <math>F.</math> If the area of triangle <math>AEF</math> is <math>15,</math> find the area of <math>ABCD</math>.
  
 
==Solution==
 
==Solution==
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We first find the chance of the pig spelling "carrot" correctly in one try.
 
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===Solution 1a===
 
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First, out of the <math>5^6</math> ways to choose the letters, only <math>\frac{6!}{2}</math> of them have the same letters as the word carrot. Then, given that the pig has chosen the words correctly, only <math>1</math> out of the <math>\frac{6!}{2}</math> ways to spell the word correctly.
 
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The probability is thus
 
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<cmath>
 
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    \frac{\frac{6!}{2}}{5^6} \cdot \frac{1}{\frac{6!}{2}} = \frac{1}{5^6}
 
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</cmath>
 
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===Solution 1b===
 
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Considering each letter position individually, it is equally likely to be any
 
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of the <math>5</math> possible letters. Thus, for each letter in carrot there is a
 
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<math>\frac{1}{5}</math> chance the pig spells the letter in that position correctly.
 
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The answer is thus <math>\frac{1}{5^6}</math>.
 
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Now let <math>x</math> be the expected number of turns required for the pig to
 
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guess correctly.
 
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We have that
 
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<cmath>
 
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    x = 1 + \frac{5^6 - 1}{5^6} \cdot x
 
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</cmath>
 
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which implies that <math>x = \boxed{15625}</math>
 

Latest revision as of 19:14, 2 May 2025

Problem

Let $ABCD$ be a parallelogram such that $E$ is a point on $CD$ such that $\frac{CE}{DE}=\frac{2}{3}.$ Suppose that $BE$ and $AC$ intersect at $F.$ If the area of triangle $AEF$ is $15,$ find the area of $ABCD$.

Solution

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