BdMO National 2012: Higher Secondary 10

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Moon
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BdMO National 2012: Higher Secondary 10

Unread post by Moon » Sat Feb 11, 2012 11:23 pm

Problem 10:
Consider a function $f: \mathbb{N}_0\to \mathbb{N}_0$ following the relations:
  • $f(0)=0$
  • $f(np)=f(n)$
  • $f(n)=n+f\left ( \left \lfloor \dfrac{n}{p} \right \rfloor \right)$ when $n$ is not divisible by $p$
Here $p > 1$ is a positive integer, $\mathbb{N}_0$ is the set of all nonnegative integers and $\lfloor x \rfloor$ is the largest integer smaller or equal to $x$.
Let, $a_k$ be the maximum value of $f (n)$ for $0\leq n \leq p^k$. Find $a_k$.
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Moon
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Re: BdMO National 2012: Higher Secondary 10

Unread post by Moon » Sun Feb 12, 2012 9:22 am

You want to maximize your "digits" in the base $p$ representation of $n$. :D
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sourav das
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Re: BdMO National 2012: Higher Secondary 10

Unread post by sourav das » Tue Feb 21, 2012 1:30 pm

Corei13 used Moon bhaia's method (A direct killing method hi, hi, hi). A different way hint (My contest solution way) (A soft way killing method, ha ha ha)
Use induction to prove $a_k= f(p^k-1)$
You spin my head right round right round,
When you go down, when you go down down......
(-$from$ "$THE$ $UGLY$ $TRUTH$" )

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*Mahi*
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Re: BdMO National 2012: Higher Secondary 10

Unread post by *Mahi* » Tue Feb 21, 2012 2:23 pm

sourav das wrote:Corei13 used Moon bhaia's method (A direct killing method hi, hi, hi). A different way hint (My contest solution way) (A soft way killing method, ha ha ha)
Use induction to prove $a_k= f(p^k-1)$
I did the same as Corei13/moon bhai in both H.Sec 9 and 10 :)
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