Search found 69 matches
- Sun Sep 28, 2014 10:02 pm
- Forum: Algebra
- Topic: Functional Equation( Japan final round 2008)
- Replies: 6
- Views: 5135
Re: Functional Equation( Japan final round 2008)
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: 12662
Re: অনেক বেশি বিলম্বিত পরিচিতি
I was a "বাচ্চা" back then -_-
anyway, logged in to the bdmo forum after a very long time
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: 17502
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: 5769
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
- Forum: International Mathematical Olympiad (IMO)
- Topic: IMO MOCK-5(ii)
- Replies: 6
- Views: 5769
IMO MOCK-5(ii)
Determine with proof all functions $f:\mathbb{R}_{0} \rightarrow \mathbb{R}_{0}$ such that
$4f(x) \geq 3x$ and $f(4f(x)-3x)=x$ $\forall x \in \mathbb{R}_{0}$
$4f(x) \geq 3x$ 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: 2104
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:
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: 2209
Re: Generalized inequality
Direct application of Jensen.
Use $\phi(x) = ln(sinx)$ that implies $\phi''(x) < 0$. So $\phi$ is concave.
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: 1634
Putnam 2010 B4
Find all pairs of polynomials $p(x)$ and $q(x)$ with real coefficients for which
$p(x)q(x+1)-p(x+1)q(x)=1.$
$p(x)q(x+1)-p(x+1)q(x)=1.$
- Tue Jun 19, 2012 12:52 pm
- Forum: Algebra
- Topic: FE/ DE/ IE :P
- Replies: 0
- Views: 1483
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)
$\left ( \int f(x) dx \right )\left ( \int \frac{1}{f(x)} dx \right ) = -1$
(might be easy)
- Tue Jun 12, 2012 11:11 pm
- Forum: Number Theory
- Topic: IMOSL-2010 N2
- Replies: 1
- Views: 2084
IMOSL-2010 N2
Find all pairs $(m , n)$ of nonnegative integers for which
$m^{2} + 2 \cdot 3^{n} = m(2^{n + 1} - 1)$
$m^{2} + 2 \cdot 3^{n} = m(2^{n + 1} - 1)$