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Prove that if $x_1,x_2,\ldots,x_n>0$, then $(1+x_1)(1+x_1+x_2)\ldots(1+x_1+x_2+\ldots+x_n) \ge \sqrt{(n+1)^{n+1}}\sqrt{x_1x_2\ldots x_n}$.

inequality asked Dec 22, 2013 at 15:01

math.stackexchange.com

12 votes

Find all $x,y,z\in\mathbb N$, $x,y,z>1$ such that satisfy $x\mid yz+1$, $y\mid xz+1$, and $z\mid xy+1$

elementary-number-theory divisibility asked Jan 17, 2014 at 21:09

math.stackexchange.com

11 votes

If $a^4+b^4\in\mathbb Q$ and $a^3+b^3\in\mathbb Q$ and $a^2+b^2\in\mathbb Q$, prove that $a+b\in\mathbb Q$ and $ab\in\mathbb Q$.

elementary-number-theory contest-math factoring rational-numbers asked May 22, 2014 at 12:53

math.stackexchange.com

9 votes

No confirmation pop up when deleting comments (Android app)

bug status-completed comments android-app asked Feb 8, 2015 at 4:53

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9 votes

Prove that there are no such different positive numbers that satisfy both $a+b=c+d$ and $a^3+b^3=c^3+d^3$.

elementary-number-theory asked Dec 12, 2013 at 18:23

math.stackexchange.com

9 votes

If $x,y,z\in\mathbb R\setminus \{1\}$ and $xyz=1$, prove that $\frac{x^2}{(x-1)^2}+\frac{y^2}{(y-1)^2}+\frac{z^2}{(z-1)^2}\ge 1$.

inequality asked Dec 22, 2013 at 23:45

math.stackexchange.com

8 votes

If $n\in\mathbb N$ and $4^n+2^n+1$ is prime, prove that there exists an $m\in\mathbb N\cup\{0\}$ such that $n=3^m$.

elementary-number-theory divisibility asked Dec 28, 2013 at 0:52

math.stackexchange.com

8 votes

$a,b,n,d\in \mathbb N$. $a,b,d$ are different numbers from the interval $(n^2;n^2+n)$. Prove that it can't be true that $d|ab$.

elementary-number-theory asked Dec 22, 2013 at 17:07

math.stackexchange.com

7 votes

Prove that $\frac{x_1}{1+x_2+x_3+\ldots+x_n}+\frac{x_2}{1+x_1+x_3+\ldots+x_n}+\ldots+\frac{x_n}{1+x_1+x_2+\ldots+x_{n-1}}\ge\frac{n}{2n-1}$.

inequality asked Dec 22, 2013 at 19:04

math.stackexchange.com

7 votes

$\forall x\in\mathbb R$, $|x|\neq 1$ it is known that $f\left(\frac{x-3}{x+1}\right)+f\left(\frac{3+x}{1-x}\right)=x$. Find $f(x)$.

functional-equations asked Mar 4, 2014 at 19:26

math.stackexchange.com

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user26486

Prove that if $x_1,x_2,\ldots,x_n>0$, then $(1+x_1)(1+x_1+x_2)\ldots(1+x_1+x_2+\ldots+x_n) \ge \sqrt{(n+1)^{n+1}}\sqrt{x_1x_2\ldots x_n}$.

Find all $x,y,z\in\mathbb N$, $x,y,z>1$ such that satisfy $x\mid yz+1$, $y\mid xz+1$, and $z\mid xy+1$

If $a^4+b^4\in\mathbb Q$ and $a^3+b^3\in\mathbb Q$ and $a^2+b^2\in\mathbb Q$, prove that $a+b\in\mathbb Q$ and $ab\in\mathbb Q$.

No confirmation pop up when deleting comments (Android app)

Prove that there are no such different positive numbers that satisfy both $a+b=c+d$ and $a^3+b^3=c^3+d^3$.

If $x,y,z\in\mathbb R\setminus \{1\}$ and $xyz=1$, prove that $\frac{x^2}{(x-1)^2}+\frac{y^2}{(y-1)^2}+\frac{z^2}{(z-1)^2}\ge 1$.

If $n\in\mathbb N$ and $4^n+2^n+1$ is prime, prove that there exists an $m\in\mathbb N\cup\{0\}$ such that $n=3^m$.

$a,b,n,d\in \mathbb N$. $a,b,d$ are different numbers from the interval $(n^2;n^2+n)$. Prove that it can't be true that $d|ab$.

Prove that $\frac{x_1}{1+x_2+x_3+\ldots+x_n}+\frac{x_2}{1+x_1+x_3+\ldots+x_n}+\ldots+\frac{x_n}{1+x_1+x_2+\ldots+x_{n-1}}\ge\frac{n}{2n-1}$.

$\forall x\in\mathbb R$, $|x|\neq 1$ it is known that $f\left(\frac{x-3}{x+1}\right)+f\left(\frac{3+x}{1-x}\right)=x$. Find $f(x)$.