Search found 155 matches

by sowmitra
Tue Oct 14, 2014 9:00 pm
Forum: Combinatorics
Topic: Solution to a Non-Linear Recurrence
Replies: 4
Views: 1312

Re: Solution to a Non-Linear Recurrence

Any help... :?
by sowmitra
Mon Oct 13, 2014 3:11 pm
Forum: Combinatorics
Topic: Solution to a Non-Linear Recurrence
Replies: 4
Views: 1312

Re: Solution to a Non-Linear Recurrence

To tell the truth, it's a product of my thought... I came up with this recurrence while trying to find the Equivalent Resistance of an Infinite Fractal-Circuit that I made up. :roll: The original recurrence I was trying to solve had $c=3, c_1=4, c_2=-2, c_3=-2$ and $a_0=\frac{2}{3}$. I was wondering...
by sowmitra
Sun Oct 12, 2014 8:32 pm
Forum: Combinatorics
Topic: Solution to a Non-Linear Recurrence
Replies: 4
Views: 1312

Solution to a Non-Linear Recurrence

Let $(a_n)$ be a sequence of numbers which satisfies the following recurrence relation for all $n\geq 1$, \[c\cdot a_na_{n-1} + c_1\cdot a_n + c_2\cdot a_{n-1} + c_3=0 \] where, $c_1, c_2, c_3$ and $c$ are constants, $c\neq 0$, and $a_0\neq-\frac{c_1}{c}$ is given. Does a general solution in CLOSED ...
by sowmitra
Fri Oct 03, 2014 11:04 pm
Forum: Secondary Level
Topic: Directed Angle
Replies: 3
Views: 1251

Re: Directed Angle

I think this might help...
Courtesy of R. A. Johnson, MAA, and, JSTOR :
http://www.jstor.org/stable/2973001
by sowmitra
Tue Aug 26, 2014 1:09 am
Forum: Physics
Topic: Resultant Resistance
Replies: 2
Views: 1127

Re: Resultant Resistance

This might help...It has some wonderful uses of symmetries in Electric Circuits:
Resistors Cube Problems :)
by sowmitra
Thu Aug 21, 2014 6:40 pm
Forum: Algebra
Topic: Proofathon Inequality
Replies: 4
Views: 1160

Re: Proofathon Inequality

Straightforward Jensen... Let, $f(x)=\sqrt{1+x^2}\Rightarrow f''(x)=(1+x^2)^{-\frac{3}{2}}> 0 \; \forall x\in \mathbb{R}$ $\therefore f(x)$ is convex. So, \[\frac{\sum_{k=1}^n f(k)}{n}\geq f\left(\frac{\sum_{k=1}^n k}{n}\right)\] \[\Rightarrow \sum_{k=1}^n\sqrt{k^2+1}\geq n\cdot f\left(\frac{n+1}{2}...
by sowmitra
Thu Feb 13, 2014 7:50 pm
Forum: Social Lounge
Topic: Best of Luck :)
Replies: 2
Views: 1458

Best of Luck :)

Best of luck to all the participants of BDMO 2014. Sleep well, keep your head clear and give it your best... :D
by sowmitra
Thu Feb 13, 2014 7:31 pm
Forum: National Math Olympiad (BdMO)
Topic: warm-up problems for national BdMO'14
Replies: 25
Views: 4045

Re: warm-up problems for national BdMO'14

Solution to Problem 12: Lemma: If $G$ is the centroid of $\triangle ABC$, $E, F$ the mid-points of $AB, AC$, and, $AG \cap EF=S$, then, $GS:SA=1:3$ and $S$ is the mid-point of $EF$. Proof of Lemma: If $M$ is the mid-point of $BC$, then, $G$ is also the centroid of $\triangle MEF$. $\therefore GS=2G...
by sowmitra
Thu Feb 13, 2014 6:30 pm
Forum: Geometry
Topic: Touching Circumcircles around Incentre [Self-Made]
Replies: 4
Views: 993

Re: Touching Circumcircles around Incentre [Self-Made]

Nice solution :)
It was inspired from JBMO 1997 #3.
by sowmitra
Wed Feb 12, 2014 2:58 pm
Forum: Geometry
Topic: Touching Circumcircles around Incentre [Self-Made]
Replies: 4
Views: 993

Re: Touching Circumcircles around Incentre [Self-Made]

Inspired from a JBMO problem.