## Search found 153 matches

Thu Feb 23, 2012 10:29 am
Forum: Algebra
Topic: Continuous function: Q <--> R\Q
Replies: 9
Views: 2685

### Re: Continuous function: Q <--> R\Q

if co-domain of $g(x)=x+f(x)$ have two distinct element $a,b$ then it's co-domain also have a rational number between $a$ and $b$ by mean value theorem as $g$ is continuous. so, $g$ is constant and $f(x)=c-x$ where $c$ is a irrational number. But then $f(\frac{c}{2})=\frac{c}{2}$ which is a contradi...
Mon Feb 13, 2012 8:40 pm
Forum: Combinatorics
Topic: Probability: coin toss
Replies: 7
Views: 3106

### Re: Probability: coin toss

Let $C_n$ is the number of such sequence, Let, in a such sequence there are $k$ Heads $a_0-1$ is the number of Tails before the first Head $a_i$ is the number of Tails between the $(i-1)$'th and $i$'th Head, $0<i<k$ $a_k-1$ is the number of Tails after the $k$'th Head It's easy to see that, $a_i>0$ ...
Wed Feb 01, 2012 9:31 pm
Topic: A proposed problem of IMO
Replies: 12
Views: 4653

### Re: A proposed problem of IMO

@Adib: Your Induction hypothesis is not enough to prove the problem, it just proves a case of the given problem. Note that, the problem asked you to prove for all $c_i$ with only one condition $(n-1)\sum_{i\le n}{c_i^2} = \left ( \sum_{i\le n}{c_i} \right )^2$. That is, you can't add a extra constra...
Tue Jan 31, 2012 11:02 pm
Forum: Number Theory
Topic: Sequences of interest
Replies: 5
Views: 2013

Yes! Let, $[x]$ denotes $x$ rounded to closest integer. Then it is easy to find, $[x] = \lfloor x + \frac{1}{2} \rfloor$ And, another obvious thing is, $\lceil x \rceil = \lfloor x \rfloor +1$ iff $x \not\in \mathbb{Z}$ It can also be easily found that, $\left[\sqrt{n}\right]=\left[\sqrt{n-\frac{1}{... Thu Jan 26, 2012 3:17 pm Forum: Number Theory Topic: Sequences of interest Replies: 5 Views: 2013 ### Re: Sequences of interest $G(n) = \left\lceil n-\frac{1}{2}+\sqrt{n-\frac{1}{2}}\right\rceil$ Wed Jan 25, 2012 10:29 pm Forum: Combinatorics Topic: Digit-swapping game (own) Replies: 5 Views: 2638 ### Re: Digit-swapping game (own) This can be generalized for all natural number$n$. Let, for any permutation$(a_1,a_2,\cdots,a_n)$of$(0,2,\cdots, n-1)$,$f_n(a_1,a_2,\cdots,a_n) = \sum_{n > i \ge 0} {\left|\{\ j \ |\ j<i, a_j < a_i\ \}\right|}$Then, it can be ( not so maybe ) easily verified that,$f_n(\cdots,a_{i-1},a_i,a_{i+...
Mon Jan 09, 2012 8:08 pm
Forum: Computer Science
Topic: factorial
Replies: 4
Views: 2585

### Re: factorial

You can input a large number as a string and store it in a array. It'll be helpfull if you know struct (just Google) Here's a sample elementary code ( in C++, I don't know C that much :? ) : #define MAXS something // 'something' is the maximum number of digits you need struct BigInt { int digit[MAXS...
Sat Jan 07, 2012 1:03 pm
Forum: Asian Pacific Math Olympiad (APMO)
Topic: APMO 2011 problem-5
Replies: 5
Views: 2657

### Re: APMO 2011 problem-5

I asked why $f(2xf(y))-2xf(y)=f(xf(y))-f(y)f(x) \leq 0$? and why $x\longrightarrow 2x\Longrightarrow f(xf(y)) \leq xf(y)$ ?
Sat Jan 07, 2012 3:03 am
Forum: Asian Pacific Math Olympiad (APMO)
Topic: APMO 2011 problem-5
Replies: 5
Views: 2657

### Re: APMO 2011 problem-5

*Mahi* wrote: $f(2xf(y))-2xf(y)=f(xf(y))-f(y)f(x) \leq 0$
*Mahi* wrote: So, setting $x$ instead of $2x$, $f(xf(y)) \leq xf(y)$, and thus $yf(x) \geq f(xy)$.
Why ? :-/
Thu Jan 05, 2012 2:43 pm
Forum: Computer Science
Topic: প্রোগ্রামিং ক্যাম্প
Replies: 2
Views: 1647