## Search found 69 matches

Sun Sep 28, 2014 10:02 pm
Forum: Algebra
Topic: Functional Equation( Japan final round 2008)
Replies: 6
Views: 2525

Let us denote the given statement by $P(x,y)$. Then, $P(x,f(x))\Rightarrow f(x+f(x))f(0)=xf(x)-f(x)f(f(x))$. $P(x,0)\Rightarrow f(x)f(f(x))=xf(x)$. So, $f(x+f(x))f(0)=0 \forall x\in \mathbb{R}$. So we have two cases here. $\textbf{Case 1:}$ When $f(x+f(x))=0\forall x\in \mathbb{R}$. $P(x+f(x),-x)\R... Sun Sep 07, 2014 4:29 pm Forum: Introductions Topic: অনেক বেশি বিলম্বিত পরিচিতি Replies: 10 Views: 7953 ### Re: অনেক বেশি বিলম্বিত পরিচিতি I was a "বাচ্চা" back then -_- anyway, logged in to the bdmo forum after a very long time Fri Nov 09, 2012 11:17 pm Forum: News / Announcements Topic: Active users for marathon Replies: 23 Views: 8539 ### Re: Active users for marathon আমিও আছি অবশ্যই Wed Oct 17, 2012 10:58 am Forum: International Mathematical Olympiad (IMO) Topic: IMO MOCK-5(ii) Replies: 6 Views: 3205 ### Re: IMO MOCK-5(ii) here's my solution! The solution is$f(x)=x$;$\forall x \in \mathbb{R}_{0}P(4f(x)-3x) \Rightarrow f(4f(4f(x)-3x)-3(4f(x)-3x))=4f(x)-3x and f(4f(4f(x)-3x)-3(4f(x)-3x)) \geq 0\Rightarrow f(13x-12f(x))=4f(x)-3x and 13x-12f(x) \geq 0 \cdot \cdot \cdot (i)$now define$g:\mathbb{R}_{0} \rightarro...
Wed Oct 17, 2012 7:55 am
Topic: IMO MOCK-5(ii)
Replies: 6
Views: 3205

### IMO MOCK-5(ii)

Determine with proof all functions $f:\mathbb{R}_{0} \rightarrow \mathbb{R}_{0}$ such that
$4f(x) \geq 3x$ and $f(4f(x)-3x)=x$ $\forall x \in \mathbb{R}_{0}$
Tue Jul 31, 2012 10:37 pm
Forum: Algebra
Topic: A Generalized Inequality
Replies: 1
Views: 1028

### A Generalized Inequality

Prove that...
the inequality $\sum_{sym}\frac{a}{\sqrt{a^2 + kbc}} \geq \frac{3}{\sqrt{1+k}}$
is true for all $a, b, c > 0$ $iff$ $k \geq 8$.

comment:
Tue Jul 31, 2012 10:07 pm
Forum: Algebra
Topic: Generalized inequality
Replies: 2
Views: 1112

### Re: Generalized inequality

Direct application of Jensen.
Use $\phi(x) = ln(sinx)$ that implies $\phi''(x) < 0$. So $\phi$ is concave.
Wed Jun 27, 2012 6:38 pm
Forum: Algebra
Topic: Putnam 2010 B4
Replies: 0
Views: 845

### Putnam 2010 B4

Find all pairs of polynomials $p(x)$ and $q(x)$ with real coefficients for which
$p(x)q(x+1)-p(x+1)q(x)=1.$
Tue Jun 19, 2012 12:52 pm
Forum: Algebra
Topic: FE/ DE/ IE :P
Replies: 0
Views: 701

### FE/ DE/ IE :P

Find all functions $f$ such that
$\left ( \int f(x) dx \right )\left ( \int \frac{1}{f(x)} dx \right ) = -1$

(might be easy)
Tue Jun 12, 2012 11:11 pm
Forum: Number Theory
Topic: IMOSL-2010 N2
Replies: 1
Views: 1105

### IMOSL-2010 N2

Find all pairs $(m , n)$ of nonnegative integers for which
$m^{2} + 2 \cdot 3^{n} = m(2^{n + 1} - 1)$