Suppose f is a function such that \( f:\mathbb{Z} ^+\to \mathbb{Z} ^+ \) satisfying

\(f(1)=1, f(2n)=f(n)\) and \(f(2n+1)=f(2n)+1\),

for all positive integers \(n\).

Find the maximum of \(f(n)\) when \( 1 \leq n \leq 2013\).

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## Comments

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TopNewest\(f(n)\) is the number of \(1\)s in the binary expansion of \(n\). Since \(2^{11} = 2048\), the smallest integer \(n\) with \(f(n) = 11\) is \(2047\). Since \(f(1983) = 10\), the maximum of \(f(n)\) over \(1 \le n \le 2013\) is \(10\).

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This function doubtlessly suffices the condition, & the solution is beautiful. But is this definition of \(f(n)\) unique? Why?

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Each number can be constructed, according to its binary expansion, by applying these rules.

We start with f(1), or "1" in binary. The f(2n) equation will tell us what f provides when we append 0 to our input, that is, f(10), and f(2n+1) tells us what happens when we append a 1.

e.g. f(110101) is computed by f(1) --> f(11) --> f(110) --> f(1101) --> f(11010) --> f(110101).

(all numbers in binary)

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Prove it by (strong) induction on \(n\).

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Please help me solve my problem above!

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Beat me to it... although it has been some time since it was posted.

But, yes, although it would have sufficed simply to use \(2^{10}-1\) (since there cannot be any more \(1\)s, as this number would then be \(2^{11}-1\)).

The argument for the sum of the digits of the binary expansion can be seen by simple construction

(this is just a quick, non-rigorous sketch)of a two-case inductive proof:Allow the number \(n\) to be of the form \(n=a_k2^{k-1}+a_{k-1}2^{k-2}+\cdots+a_1\), where \(a_i\in\{0,1\}\) and \(a_k=1\).

Case 1Then, if this number is odd, we must have \(a_1=1\), hence, we have \(f(2m+1)=f(2m)+1\) for some \(m\in\mathbb{Z}\). So, now our number becomes: \[ 2m+1=a_k2^{k-1}+a_{k-1}2^{k-2}+\cdots+a_22^{1}+1 \] \[ m=a_k2^{k-2}+a_{k-1}2^{k-3}+\cdots+a_2 \] This adds unity to our count, removes the last digit, and continues the process until the only value left of \(m\) is one.

Case 2Say that \(n\) is even, then we must have \(a_1=0\). Clearly, then, we have: \(f(2m)=f(m)\), which, after dividing, gives us a new number with the last digit removed and adds nothing to the count (as the last digit is 0 in base 2).

The same process applies.

Hence, we are done (note that this proof can easily be extended for uniqueness of definition as Paramjit S.'s request).

Hopefully this makes the case a little more clear for those in doubt.

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a very good thing indeed.good work Mark H.

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Hi there! Seeing that I am having trouble with starting a discussion, I decided to just post it as a comment instead.

While preparing for the South African National Olympiad, the following problem featured in my training material:

"An 11 sided polygon has its vertices on a circle. How many triangles can be formed using three vertices of the polygon but sharing no side with a side of the polygon?"

Can somebody please provide a solution?

Thanks!

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1009!

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Sorry, it's not 1009 factorial. :) Lol Careful with the !

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Is it 11?

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