$x$ be a real number and $f(x)$ be a real-valued function on $x$ such that $f(x)$ is:
Letthe next palindrome larger than $x$ (if $x$ is not a palindrome); or
$x$ (if $x$ is a palindrome).
For example, $f(1001)=1001$ , $f(1001.01)=f(1002)=1111$ .
Let
$A=\int _{ 999 }^{ 2015 }{ f(x) \mbox{ } dx }$
$B=f(...(f(f(1603)-58)-58)...)-58$
Find the value of $\frac{A}{B}$ .
This problem is part of the set 2015 Countdown Problems .
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f(1603)=1661. 1661- 58 =1603. So we get back to where we were. Iterating any number of times will get us back to 1603. So B=1603. Now for 999<x<=1001 we have f(x)= 1001. 1001<x<=1111 f(x)= 1111. Continuing in this fashion gives us f(x)= 2(1001) + 110 ( 1111+1221+ .... + 1991) + 11(2002) +13(2112) = 1586970 = A.
So A/B = 990