Search found 64 matches

by Atonu Roy Chowdhury
Thu Dec 03, 2020 11:35 pm
Forum: Algebra
Topic: Functional equation
Replies: 1
Views: 877

Re: Functional equation

The equation $f(x+y)=f(x)+f(y)$ is generally called Cauchy Functional Equation . Sadly, we can't find an explicit solution for Cauchy Functional Equation when the domain is Real (if no other conditions are given). However we can find solutions when the domain is Natural numbers or Rational numbers. ...
by Atonu Roy Chowdhury
Wed Jun 27, 2018 8:14 pm
Forum: Social Lounge
Topic: short query
Replies: 9
Views: 8117

Re: short query

Neelkhet may help you to get math books.
by Atonu Roy Chowdhury
Mon Apr 23, 2018 9:59 pm
Forum: Algebra
Topic: FE from USAMO 2002
Replies: 4
Views: 5117

Re: FE from USAMO 2002

The fact that $f(x^2)=xf(x)$ implies that $f(x^2-y^2)=f(x^2)-f(y^2)$ so it implies that $f(a-b)=f(a)-f(b)$ for $a$ and $b$ positive perfect square; why do you say that this expression is true also for $a$ and $b$ positive numbers (and so no necessary positive perfect squares...)? And why after do y...
by Atonu Roy Chowdhury
Sat Apr 21, 2018 11:25 pm
Forum: Algebra
Topic: Inequality with a,b,c sides of a triangle
Replies: 7
Views: 7204

Re: Inequality with a,b,c sides of a triangle

Katy729 wrote:
Sat Apr 21, 2018 10:53 pm
Yes, now understand. Very gentle Atonu! :)
Okay, you can check this out if you have confusion about the proof of negative exponent.
by Atonu Roy Chowdhury
Sat Apr 21, 2018 10:36 pm
Forum: Algebra
Topic: Inequality with a,b,c sides of a triangle
Replies: 7
Views: 7204

Re: Inequality with a,b,c sides of a triangle

Thanks Atonu, now is clear! :) Only a doubt, can you put a link where is wrote the property that you say? Because I found that the exponent ($n$) must be positive, yes in your cases the numbers are positive but where is wrote? I saw here but I didn't find the property that you say... :( https://en....
by Atonu Roy Chowdhury
Sat Apr 21, 2018 9:45 pm
Forum: Algebra
Topic: Inequality with a,b,c sides of a triangle
Replies: 7
Views: 7204

Re: Inequality with a,b,c sides of a triangle

Sorry Atonu, but I don't understand some passages... :( Why exactly \[ \sum_{cyc} \frac{\sqrt{b+c-a}}{\sqrt{b}+\sqrt{c}-\sqrt{a}}= \sum_{cyc} \sqrt{1- \frac{(x-y)(x-z)}{2x^2}} \le \sum_{cyc} (1-\frac{(x-y)(x-z)}{4x^2}) = 3 - \frac{1}{4} \sum_{cyc} x^{-2}(x-y)(x-z) \] ? And how do you use exactly Sh...
by Atonu Roy Chowdhury
Fri Apr 20, 2018 9:59 pm
Forum: Secondary Level
Topic: Easy Projective Geo
Replies: 3
Views: 4576

Re: Easy Projective Geo

Harmonic quad approach is quite intuitive. Anyone tried bash?
Btw, apart from projective I solved it by cartesian coordinates. I don't have enough patience to type that lengthy and annoying solution here.
by Atonu Roy Chowdhury
Fri Apr 20, 2018 12:20 pm
Forum: Geometry
Topic: EGMO 2018 P5
Replies: 3
Views: 10064

Re: EGMO 2018 P5

Okay, I copied it from my aops post. Let $X$ be the midpoint of the arc $AB$ not containing $C$. Also let $Y$ and $Z$ be the tangency point of $\Omega$ with $AB$ and $\Gamma$ respectively. $D$ is the feet of angle bisector of $\angle ACB$. $I$ denotes the incenter of $\triangle ABC$ Lemma 1: $X,Y,Z$...
by Atonu Roy Chowdhury
Sat Apr 14, 2018 8:21 pm
Forum: Number Theory
Topic: 2018 BDMO NT exam P4
Replies: 2
Views: 4701

Re: 2018 BDMO NT exam P4

শুভ নববর্ষ ১৪২৫ We will show that there are at least $2$ integers in the sequence. Assume the contrary, i.e. assume there are at most $1$ integer. Let $\{a_i\}_{i=1}^{n}$ be the sequence. For $m=1$, we get either $a_1$ or $a_n$ is integer. WLOG, $a_1$ is integer. For $m=2$, either $a_1+a_2$ or $a_{n...
by Atonu Roy Chowdhury
Wed Apr 11, 2018 11:47 pm
Forum: Algebra
Topic: Inequality with a,b,c sides of a triangle
Replies: 7
Views: 7204

Re: Inequality with a,b,c sides of a triangle

$\sqrt{b}+\sqrt{c}-\sqrt{a}=x,\sqrt{c}+\sqrt{a}-\sqrt{b}=y,\sqrt{a}+\sqrt{b}-\sqrt{c}=z$ $$\sum_{cyc} \frac{\sqrt{b+c-a}}{\sqrt{b}+\sqrt{c}-\sqrt{a}} = \sum_{cyc} \sqrt{1- \frac{(x-y)(x-z)}{2x^2}} \le \sum_{cyc} 1-\frac{(x-y)(x-z)}{4x^2} = 3 - \frac{1}{4} \sum_{cyc} x^{-2}(x-y)(x-z) $$ So, we need t...