1975 Canadian MO Problems/Problem 1

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Problem 1

Simplify \[\left(\frac{1\cdot2\cdot4+2\cdot4\cdot8+\cdots+n\cdot2n\cdot4n}{1\cdot3\cdot9+2\cdot6\cdot18+\cdots+n\cdot3n\cdot9n}\right)^{1/3}\].

Solution

$\left(\frac{1\cdot2\cdot4+2\cdot4\cdot8+\cdots+n\cdot2n\cdot4n}{1\cdot3\cdot9+2\cdot6\cdot18+\cdots+n\cdot3n\cdot9n}\right)^{1/3}$


$\left(\frac{2^3\cdot1+2^3\cdot2^3+2^3\cdot3^3+2^3\cdot4^3+\cdots+2^3n^3}{3^3+3^3\cdot2^3+3^3\cdot3^3+3^3\cdot4^3+\cdots+3^3n^3}\right)^{1/3}$


$\left[\frac{2^3\cancel{(1^3+2^3+3^3+\cdots+n^3)}}{3^3\cancel{(1^3+2^3+3^3+\cdots+n^3)}}\right]^{1/3}$


$\boxed{\frac{2}{3}}$


1975 Canadian MO (Problems)
Preceded by
First question
1 2 3 4 5 6 7 8 9 10 Followed by
Problem 2


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