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Let $f$ be a function from the positive integers to the positive integers that satisfies the property: $$ f(x+y)=f(x)f(y) $$ for all pairs of positive integers $(x,y)$.

If we are given that $f(2)=9$, what is the value of $f(5)$?

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    $\begingroup$ Nice try, what have you tried? $\endgroup$ – Michael Hoppe Oct 17 '13 at 14:03
  • $\begingroup$ What is $f(1)$ under these conditions? $\endgroup$ – Daniel Fischer Oct 17 '13 at 14:05
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    $\begingroup$ This is a live math problem on Brilliant. Please close this for a week. - Calvin Lin, Brilliant Challenge Master $\endgroup$ – Calvin Lin Oct 17 '13 at 15:04
  • $\begingroup$ I flagged it. Lets see what happens. $\endgroup$ – Marc Palm Oct 17 '13 at 15:36
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    $\begingroup$ This question is a question on an open contest and the contest organisation has requested that it be closed until the contest finishes. $\endgroup$ – Peter Taylor Oct 17 '13 at 16:12
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$f(2) = f(1)^2 = 9$, what is $f(5) = f(1)^{5}$ then?

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    $\begingroup$ This is a live math problem on Brilliant. Please close this for a week. - Calvin Lin, Brilliant Challenge Master $\endgroup$ – Calvin Lin Oct 17 '13 at 15:06
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We have $$ f(2)=f(1)^2=9. $$ Thus $f(1)=3$ and $$ f(5)=f(1)^5=3^5=243. $$

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  • $\begingroup$ This is a live math problem on Brilliant. Please close this for a week. - Calvin Lin, Brilliant Challenge Master $\endgroup$ – Calvin Lin Oct 17 '13 at 15:06
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As mentioned, this is a live math problem on Brilliant. Solutions are available. The image intentionally cuts off the first solution.

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