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For all positive integer $x$, $f(f(x)) = 4x+3$; and for ONLY ONE positive integer $k$, $f(5^k) = 5^k \times 2^{k-2} + 2^{k-3}$. Find $f(2015)$.

Dont know where to start. Any hint will be helpful. Dont give full solution.
Source: BdMO 2016 Dhaka regional Higher Secondary.

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    what is $f$? what is its domain and codomain? – 2017-01-06
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    This is the exact problem statement. The main function is for all positive number – 2017-01-06
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    have you been able to find a function that works (trial and error works easily)? at least that way you'll be able to find the result. – 2017-01-06

2 Answers 2

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Sort of a hint: Since $f(f(x))$ is linear, I suspect that so is $f(x)$. I assumed that $f(x) = ax+b$ and was able to get the answer.

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    $f(x) = 2x+1$? I get this. – 2017-01-06
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    yes, that one works, of course the "tought" part of the problem is proving that any function that works must have $f(2015)=4031$. Although this will probably earn you $2$ points out of $7$. – 2017-01-06
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    And there are some other proofs that I need to show. like there is only one integer $k$ for which the second equation works. And there might be some small thing. After that I will have at-least 5/7 ? – 2017-01-07
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Hint 1: f(x) is injective

Hint 2:$2017=503\cdot 4+3,503=4\cdot 125+3$