Search found 69 matches

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

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...
by Sazid Akhter Turzo
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 :D
by Sazid Akhter Turzo
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 :D
by Sazid Akhter Turzo
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...
by Sazid Akhter Turzo
Wed Oct 17, 2012 7:55 am
Forum: International Mathematical Olympiad (IMO)
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 $f(4f(x)-3x)=x$ $\forall x \in \mathbb{R}_{0}$
by Sazid Akhter Turzo
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:
It's a generalized form of IMO SL-2001-A6 ;)
by Sazid Akhter Turzo
Tue Jul 31, 2012 10:07 pm
Forum: Algebra
Topic: Generalized inequality
Replies: 2
Views: 1112

Re: Generalized inequality

Direct application of Jensen. :D
Use $\phi(x) = ln(sinx)$ that implies $\phi''(x) < 0$. So $\phi$ is concave.
by Sazid Akhter Turzo
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
$p(x)q(x+1)-p(x+1)q(x)=1.$
by Sazid Akhter Turzo
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)
by Sazid Akhter Turzo
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
$m^{2} + 2 \cdot 3^{n} = m(2^{n + 1} - 1)$