Actually it should be " We can reduce $1,2,\cdots ,n$ to $1, 2, \cdots \left \lfloor \frac{n}{2} \right \rfloor$ "
Otherwise you won't do it in $ \lfloor log_2 {n} \rfloor + 1$ steps in all cases.
Search found 153 matches
- Thu Oct 13, 2011 5:48 pm
- Forum: Combinatorics
- Topic: COMBINATORICS MARATHON: SEASON 2
- Replies: 22
- Views: 13410
- Wed Oct 12, 2011 7:50 pm
- Forum: Combinatorics
- Topic: COMBINATORICS MARATHON: SEASON 2
- Replies: 22
- Views: 13410
Re: COMBINATORICS MARATHON: SEASON 2
Oops! I forgot it is a marathon ! New problem: You are given $n$ integers $1,2, \cdots ,n$. At each step you can choose any number of numbers from the $n$ numbers and then choose any positive integer $d$. Then you subtract $d$ from all of your chosen numbers. Your task is to make all the numbers equ...
- Tue Oct 11, 2011 11:20 pm
- Forum: Combinatorics
- Topic: COMBINATORICS MARATHON: SEASON 2
- Replies: 22
- Views: 13410
Re: COMBINATORICS MARATHON: SEASON 2
Let $a_n$ is the solution. Let, $b_n$ is the numbers of tiling WITH considering reflections. Then, $b_{n} = b_{n-1} +2 b_{n-2}$ Let, $c_n$ is the numbers of tiling which remains same after reflection. Then, $c_{2n} = b_n + 2*b_{n-1}$ and $c_{2n+1} =b_n$ Now, $b_n$ = Total number of tiling = symmetri...
- Tue Oct 11, 2011 3:37 pm
- Forum: Combinatorics
- Topic: COMBINATORICS MARATHON: SEASON 2
- Replies: 22
- Views: 13410
Re: COMBINATORICS MARATHON: SEASON 2
I just found two simple recursions. Those recursions can be solved either using generating functions or using other algebra stuffs.
Re: need help
Because it's so trivial! Checking for a constant function solution is a trivial attempt. You say, "If $f$ is constant, then $f(x) =c$ for all $x$. So, $c + c^2 = 3c$, and so, $c=0$ or $2$." Then you say, "So, now let $f$ isn't a constant function." So you only have to worry about non-trivial solutio...
Re: need help
$\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q}$. So any function that satisfies a relation in $\mathbb{Q}$ also satisfies the relation in $\mathbb{Z},\mathbb{N}$.
I couldn't understand your second question :-/
I couldn't understand your second question :-/
- Wed Sep 21, 2011 10:55 pm
- Forum: Algebra
- Topic: Nice Problem ( Inspired from Nayel Vai )
- Replies: 1
- Views: 1773
- Wed Sep 21, 2011 10:51 pm
- Forum: Algebra
- Topic: Nice Problem ( Inspired from Nayel Vai )
- Replies: 1
- Views: 1773
Nice Problem ( Inspired from Nayel Vai )
Prove that, there is no positive real $c$ such that $\{nr\} > c$ for all $n \in \mathbb{Z}/\{0\}$. Where $r$ is an irrational number.
- Wed Sep 21, 2011 10:35 pm
- Forum: Algebra
- Topic: ALGEBRA MARATHON: SEASON 2
- Replies: 30
- Views: 14946
Re: ALGEBRA MARATHON: SEASON 2
It's very easy, I found it in AoPS. Post a problem!
- Wed Sep 21, 2011 7:11 pm
- Forum: News / Announcements
- Topic: ১০০০ তম টপিক!
- Replies: 7
- Views: 5300
Re: ১০০০ তম টপিক!
১০০ তম পোস্ট, ১০০০ তম পোস্ট, ১০০০ তম টপিক --এই ট্রেবল অর্জনের পর আমার নাম প্লাটিনামক্ষরে না লিখা থাকে তাহলেই তো অন্যায়। আমি যে হীরকে সন্তুষ্ট এটাই তো অনেক।
After all, I'm the good Shepherd !
After all, I'm the good Shepherd !