## Dhaka Higher Secondary 2011/9

Problem for Higher Secondary Group from Divisional Mathematical Olympiad will be solved here.
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Please don't post problems (by starting a topic) in the "Higher Secondary: Solved" forum. This forum is only for showcasing the problems for the convenience of the users. You can post the problems in the main Divisional Math Olympiad forum. Later we shall move that topic with proper formatting, and post in the resource section.
BdMO
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### Dhaka Higher Secondary 2011/9

Consider a function $f: \mathbb N$ $\to$ $\mathbb Z$ is so defined that the following relations hold:
$f(2^n)=f(2^{n+2})\text{ and } f\left (\sum_{n\in X}^{} 2^n\right)=\sum_{n\in X}^{} f(2^n)$
where $X$ is some finite subset of $\mathbb{N} \cup \{0\}$.
Find $f(1971)$ if it is known that $f(2011) = 1$ and $f(1952) = -1$.

Avik Roy
Posts: 156
Joined: Tue Dec 07, 2010 2:07 am

### Re: Dhaka Higher Secondary 2011/9

Hint:
"Je le vois, mais je ne le crois pas!" - Georg Ferdinand Ludwig Philipp Cantor

Zzzz
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Location: 22° 48' 0" N / 89° 33' 0" E

### Re: Dhaka Higher Secondary 2011/9

Is the first relation valid for $n =0$ and the second relation valid for all $X \subset \mathbb N\cup \{0\}$ ?

*edited
Every logical solution to a problem has its own beauty.

Avik Roy
Posts: 156
Joined: Tue Dec 07, 2010 2:07 am

### Re: Dhaka Higher Secondary 2011/9

Yes, $n=0$ is allowed in the first relation as well

btw, i guess it should be mentioned that only one camper could actually 'solve' this problem during the olympiad and he found the perfect solution.
"Je le vois, mais je ne le crois pas!" - Georg Ferdinand Ludwig Philipp Cantor

Zzzz
Posts: 172
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Location: 22° 48' 0" N / 89° 33' 0" E

### Re: Dhaka Higher Secondary 2011/9

Solution:
Every logical solution to a problem has its own beauty.

TIUrmi
Posts: 61
Joined: Tue Dec 07, 2010 12:13 am
Contact:

### Re: Dhaka Higher Secondary 2011/9

Yeah the camper is Mugdho and the answer was guessed during olympiad.
"Go down deep enough into anything and you will find mathematics." ~Dean Schlicter

TIUrmi
Posts: 61
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Contact:

### Re: Dhaka Higher Secondary 2011/9

I mean he is one of them who wrote 0 in answer.
"Go down deep enough into anything and you will find mathematics." ~Dean Schlicter

Avik Roy
Posts: 156
Joined: Tue Dec 07, 2010 2:07 am

### Re: Dhaka Higher Secondary 2011/9

Mugdho did guess it and so did many others.
but I'm talking about someone solving it
"Je le vois, mais je ne le crois pas!" - Georg Ferdinand Ludwig Philipp Cantor

the arrivals
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Joined: Tue Dec 21, 2010 10:17 pm

### Re: Dhaka Higher Secondary 2011/9

whats problem with this solution??
1971=S+32+16+2+1
1995=S+64+8+2+1
2011=S+64+16+8+2+1 where S=1024+512+256+128
as f(2^n)=f(2^[n+2])
set f(S)=H(S)
so f(1971)=H(S)+2(f(1)+f(2))
and f(1995)=H(S)+2(f(1)+f(2))
so f(1995)=f(1971)=-1.....
women of purity are for men of purity and hence men of purity are for women of purity - THE HOLY QURAN

Avik Roy
Posts: 156
Joined: Tue Dec 07, 2010 2:07 am

### Re: Dhaka Higher Secondary 2011/9

@the arrivals, check the question again
"Je le vois, mais je ne le crois pas!" - Georg Ferdinand Ludwig Philipp Cantor