Difference between revisions of "1975 Canadian MO Problems/Problem 1"
(Created page with "== Problem 1 == Simplify <cmath>\left(\frac{1\cdot2\cdot+2\cdot4\cdot8+\cdots+n\cdot2n\cdot4n}{1\cdot3\cdot9+2\cdot6\cdot18+\cdots+n\cdot3n\cdot9n}\right)^{1/3}</cmath>. == S...") |
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== Problem 1 == | == Problem 1 == | ||
Simplify | Simplify | ||
| − | <cmath>\left(\frac{1\cdot2\ | + | <cmath>\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}</cmath>. |
== Solution == | == Solution == | ||
| − | + | <math>\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}</math> | |
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| + | <math>=\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}</math> | ||
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| + | <math>=\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}</math> | ||
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| + | <math>=\boxed{\frac{2}{3}}</math> | ||
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{{Old CanadaMO box|before=First question|num-a=2|year=1975}} | {{Old CanadaMO box|before=First question|num-a=2|year=1975}} | ||
![\[\left(\frac{1\cdot2\cdot4+2\cdot4\cdot8+\cdots+n\cdot2n\cdot4n}{1\cdot3\cdot9+2\cdot6\cdot18+\cdots+n\cdot3n\cdot9n}\right)^{1/3}\]](http://latex.artofproblemsolving.com/1/d/1/1d1c3ffc61429c1e6337c2b09f8682707ec2e1f2.png)
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![$=\left[\frac{2^3\cancel{(1^3+2^3+3^3+\cdots+n^3)}}{3^3\cancel{(1^3+2^3+3^3+\cdots+n^3)}}\right]^{1/3}$](http://latex.artofproblemsolving.com/c/1/0/c109733e627761f233356b79881104eb404155bb.png)
