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Difference between revisions of "1975 AHSME Problems/Problem 5"

(Created page with "The polynomial <math>(x+y)^9</math> is expanded in decreasing powers of <math>x</math>. The second and third terms have equal values when evaluated at <math>x=p</math> and <m...")
 
 
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==Problem==
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The polynomial <math>(x+y)^9</math> is expanded in decreasing powers of <math>x</math>. The second and third terms have equal values  
 
The polynomial <math>(x+y)^9</math> is expanded in decreasing powers of <math>x</math>. The second and third terms have equal values  
 
when evaluated at <math>x=p</math> and <math>y=q</math>, where <math>p</math> and <math>q</math> are positive numbers whose sum is one. What is the value of <math>p</math>?  
 
when evaluated at <math>x=p</math> and <math>y=q</math>, where <math>p</math> and <math>q</math> are positive numbers whose sum is one. What is the value of <math>p</math>?  
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The second and third term of <math>(x+y)^9</math> is <math>9x^8y</math> and <math>36x^7y^2</math>, respectively. For them to be equal when <math>x = p</math>, <math>\dfrac{p}{4} = y</math>. For them to be equal when <math>y = q</math>, <math>x = 4q</math>. Then <math>p+\dfrac{p}{4} = q+4q</math>, so <math>\dfrac{5p}{4} = 5q</math>, which simplifies to <math>p = 4q</math>. Since <math>p+q = 1</math>, <math>p = \boxed{\textbf{(B) } 4/5}</math>.
 
The second and third term of <math>(x+y)^9</math> is <math>9x^8y</math> and <math>36x^7y^2</math>, respectively. For them to be equal when <math>x = p</math>, <math>\dfrac{p}{4} = y</math>. For them to be equal when <math>y = q</math>, <math>x = 4q</math>. Then <math>p+\dfrac{p}{4} = q+4q</math>, so <math>\dfrac{5p}{4} = 5q</math>, which simplifies to <math>p = 4q</math>. Since <math>p+q = 1</math>, <math>p = \boxed{\textbf{(B) } 4/5}</math>.
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==See Also==
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{{AHSME box|year=1975|num-b=4|num-a=6}}
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{{MAA Notice}}

Latest revision as of 16:52, 19 January 2021

Problem

The polynomial $(x+y)^9$ is expanded in decreasing powers of $x$. The second and third terms have equal values when evaluated at $x=p$ and $y=q$, where $p$ and $q$ are positive numbers whose sum is one. What is the value of $p$?

$\textbf{(A)}\ 1/5 \qquad \textbf{(B)}\ 4/5 \qquad \textbf{(C)}\ 1/4 \qquad \textbf{(D)}\ 3/4 \qquad \textbf{(E)}\ 8/9$


Solution

Solution by e_power_pi_times_i


The second and third term of $(x+y)^9$ is $9x^8y$ and $36x^7y^2$, respectively. For them to be equal when $x = p$, $\dfrac{p}{4} = y$. For them to be equal when $y = q$, $x = 4q$. Then $p+\dfrac{p}{4} = q+4q$, so $\dfrac{5p}{4} = 5q$, which simplifies to $p = 4q$. Since $p+q = 1$, $p = \boxed{\textbf{(B) } 4/5}$.

See Also

1975 AHSME (Problems • Answer Key • Resources)
Preceded by
Problem 4 		Followed by
Problem 6
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30
All AHSME Problems and Solutions

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