## Search found 63 matches

- Wed Jun 27, 2018 8:14 pm
- Forum: Social Lounge
- Topic: short query
- Replies:
**9** - Views:
**7378**

### 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:
**4740**

### 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:
**6592**

- Sat Apr 21, 2018 10:36 pm
- Forum: Algebra
- Topic: Inequality with a,b,c sides of a triangle
- Replies:
**7** - Views:
**6592**

### 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:
**6592**

### 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:
**4160**

### 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:
**9629**

### 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:
**4357**

### 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:
**6592**

### 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...

### Re: FE FE FE

$f(x)f(y+k)=2f(x+(y+k)f(x))=2f(x+yf(x)+\frac{2k}{f(y)}f(x+yf(x))$ [Recall, $f(x)=\frac{2}{f(y)}f(x+yf(x)$] $\Rightarrow f(x)f(y+k) = f(x+yf(x))f(\frac{2k}{f(y)})=\frac{1}{2}f(x)f(y)f(\frac{2k}{f(y)})$ $\Rightarrow 2f(y+k)=f(y)f(\frac{2k}{f(y)})=2f(y+\frac{2k}{f(y)}f(y))=2f(y+2k)$ Inductivly, $f(y+k)...