## Search found 64 matches

- Thu Dec 03, 2020 11:35 pm
- Forum: Algebra
- Topic: Functional equation
- Replies:
**1** - Views:
**821**

### 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. ...

- Wed Jun 27, 2018 8:14 pm
- Forum: Social Lounge
- Topic: short query
- Replies:
**9** - Views:
**7950**

### Re: short query

Neelkhet may help you to get math books.

- Mon Apr 23, 2018 9:59 pm
- Forum: Algebra
- Topic: FE from USAMO 2002
- Replies:
**4** - Views:
**5016**

### 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...

- Sat Apr 21, 2018 11:25 pm
- Forum: Algebra
- Topic: Inequality with a,b,c sides of a triangle
- Replies:
**7** - Views:
**7061**

- Sat Apr 21, 2018 10:36 pm
- Forum: Algebra
- Topic: Inequality with a,b,c sides of a triangle
- Replies:
**7** - Views:
**7061**

### 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....

- Sat Apr 21, 2018 9:45 pm
- Forum: Algebra
- Topic: Inequality with a,b,c sides of a triangle
- Replies:
**7** - Views:
**7061**

### 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...

- Fri Apr 20, 2018 9:59 pm
- Forum: Secondary Level
- Topic: Easy Projective Geo
- Replies:
**3** - Views:
**4464**

### 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.

Btw, apart from projective I solved it by cartesian coordinates. I don't have enough patience to type that lengthy and annoying solution here.

- Fri Apr 20, 2018 12:20 pm
- Forum: Geometry
- Topic: EGMO 2018 P5
- Replies:
**3** - Views:
**9968**

### 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: $X,Y,Z$...

- Sat Apr 14, 2018 8:21 pm
- Forum: Number Theory
- Topic: 2018 BDMO NT exam P4
- Replies:
**2** - Views:
**4628**

### 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...

- Wed Apr 11, 2018 11:47 pm
- Forum: Algebra
- Topic: Inequality with a,b,c sides of a triangle
- Replies:
**7** - Views:
**7061**

### 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...