1975 AHSME Problems/Problem 5

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 $p+q = 1$ , $p = \boxed{\textbf{(B) } 4/5}$ .

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1975 AHSME Problems/Problem 5

Solution