Let $x,y,z>\frac 12$ satisfy $xyz=1$. Prove that
\[\frac{1}{2x-1}+\frac{1}{2y-1}+\frac{1}{2z-1}\ge 3.\]
Search found 268 matches
- Sat Mar 05, 2016 11:38 pm
- Forum: Algebra
- Topic: Inequality with xyz=1
- Replies: 1
- Views: 2961
- Sun Nov 22, 2015 8:22 pm
- Forum: Number Theory
- Topic: Factorial divisible by Mersenn Numbers
- Replies: 1
- Views: 2720
Re: Factorial divisible by Mersenn Numbers
Bang's theorem tells us that if $n\neq 1,6$ then $2^n-1$ has a primitive prime divisor $p$. If $p\le n$ then $p$ divides $2^{p-1}-1$ which is less than $2^n-1$, a contradiction. So one only needs to check $n=1,6$ of which only $n=1$ works.
- Sun Nov 22, 2015 7:29 am
- Forum: Number Theory
- Topic: #Number Theory
- Replies: 2
- Views: 4021
Re: #Number Theory
Your claim does not hold if $b=0$ so I'll assume that $ab\neq 0$. Then $p$ must be odd. Suppose that $p^2=a^2+2b^2$. Then $(p+a)(p-a)=2b^2$, so $p\pm a$ are both even. Set $p+a=2x$ and $p-a=2y$. Then $2xy=b^2$, implying that $b$ is even. Set $b=2z$. Then $xy=2z^2$. Note that $(x,y)$ divides $2p$ and...
- Sun Nov 22, 2015 7:14 am
- Forum: Number Theory
- Topic: prove it!!!
- Replies: 2
- Views: 3309
Re: prove it!!!
Let $S=\{a_1x_1+\cdots+a_nx_n:x_1,\dots,x_n\in\mathbb Z\}$. Let $d'$ be the smallest positive element of $S$. Prove the following:
(i) $d$ divides $d'$.
(ii) $d'$ divides $d$.
So $d=d'\in S$ and your conclusion will follow.
(i) $d$ divides $d'$.
(ii) $d'$ divides $d$.
So $d=d'\in S$ and your conclusion will follow.
- Sun Nov 22, 2015 7:12 am
- Forum: Number Theory
- Topic: Sequence
- Replies: 3
- Views: 4564
Re: Sequence
It's a divergent series. Neither the infinite sum nor the infinite product has a finite value. So it does not even make sense to say that they are equal. More details: https://en.wikipedia.org/wiki/Harmonic_ ... thematics)
- Wed Apr 15, 2015 12:43 am
- Forum: Higher Secondary Level
- Topic: Functional equation
- Replies: 1
- Views: 3276
Re: Functional equation
Why? What if $c=0$?joy_li wrote:$\Rightarrow -c(x-y) \neq -cx-cy$
- Sat Mar 21, 2015 7:23 pm
- Forum: Geometry
- Topic: Congruence
- Replies: 1
- Views: 3113
Re: Congruence
Hint:
- Mon Feb 23, 2015 3:55 am
- Forum: Higher Secondary Level
- Topic: Vectors around Regular Polygon
- Replies: 4
- Views: 12874
Re: Vectors around Regular Polygon
Two words:
- Sat Feb 21, 2015 1:13 am
- Forum: Algebra
- Topic: Generating Z^2
- Replies: 0
- Views: 2273
Generating Z^2
Let $a=(a_1,a_2)\in\mathbb Z^2$ where $a_1$ and $a_2$ are coprime. Show that there exists $b=(b_1,b_2)\in\mathbb Z^2$ such that any element of $\mathbb Z^2$ can be written as a $\mathbb Z$-linear combination of $a$ and $b$. Terminology: 1. A $\mathbb Z$-linear combination of $a$ and $b$ is anything ...
- Fri Feb 20, 2015 3:11 am
- Forum: Algebra
- Topic: Binomial and power of 4(or 2?)
- Replies: 3
- Views: 6121
Re: Binomial and power of 4(or 2?)
Same idea from the other thread gives a stronger bound: