## Search found 217 matches

- Thu Apr 11, 2013 9:47 pm
- Forum: Number Theory
- Topic: I Love Mr.Green
- Replies:
**5** - Views:
**2190**

### I Love Mr.Green

$a,b \in \mathbb N_0$ such that $\forall n \in \mathbb N_0$ ,$2^{n}a+b$ is a perfect square.Prove that $a=0$.

- Thu Apr 11, 2013 3:48 pm
- Forum: Number Theory
- Topic: m and n
- Replies:
**4** - Views:
**2594**

### Re: m and n

$2^m-1|2^n-1 \longleftrightarrow m|n$.Let $n=km$. As $2^m-1|\frac{2^n-1}{2^m-1}$

and $2^{m(k-1)}+2^{m(k-2)}+.............+1 \equiv k \equiv 0 (mod 2^m-1)$

\[\Longleftrightarrow 2^m-1|k\]

\[\Longleftrightarrow m(2^m-1)|n\]

and $2^{m(k-1)}+2^{m(k-2)}+.............+1 \equiv k \equiv 0 (mod 2^m-1)$

\[\Longleftrightarrow 2^m-1|k\]

\[\Longleftrightarrow m(2^m-1)|n\]

- Thu Apr 11, 2013 3:04 pm
- Forum: Number Theory
- Topic: Kiran S. Kedlaya
- Replies:
**3** - Views:
**1822**

### Kiran S. Kedlaya

Show that if $x$,$y$,$z$ are all positive integers then $(xy+1)(yz+1)(zx+1)$ is a perfect square if and only if $xy+1$,$yz+1$,$zx+1$ are all perfect squares.

- Sun Apr 07, 2013 5:18 pm
- Forum: Number Theory
- Topic: USAMO 2008: Problem 1
- Replies:
**2** - Views:
**1373**

### Re: USAMO 2008: Problem 1

$\prod k_i = a^2+a+1$ now we can take $k_n=a^2-a+1$ satisfying $(\prod k_i,k_n)=1$ and $(a^2+a+1)(a^2-a+1) =a^4+a^2+1$

- Sat Apr 06, 2013 2:04 am
- Forum: Algebra
- Topic: N'th power inequality
- Replies:
**1** - Views:
**1284**

### Re: N'th power inequality

This is not some problem i came across.It is a generalization of a problem by Mehfuz Zohir Shishir.I posted this on behalf of him.He forgot his forum password.

- Sat Apr 06, 2013 12:52 am
- Forum: Algebra
- Topic: N'th power inequality
- Replies:
**1** - Views:
**1284**

### N'th power inequality

All $a_{i}$ are positive real numbers.Prove that,

\[\sum_{cyclic} \frac{1}{a_{i}^{n}+a_{i+1}^{n}+......................+a_{i+n-2}^{n}+a_{1}a_{2}.....a_{n}} \leq \frac{1}{a_{1}a_{2}..........a_{n}}\]

\[\sum_{cyclic} \frac{1}{a_{i}^{n}+a_{i+1}^{n}+......................+a_{i+n-2}^{n}+a_{1}a_{2}.....a_{n}} \leq \frac{1}{a_{1}a_{2}..........a_{n}}\]

- Thu Mar 07, 2013 6:16 pm
- Forum: National Math Camp
- Topic: Cool but may be tough(relatively) (BOMC-2)
- Replies:
**4** - Views:
**2556**

### Re: Cool but may be tough(relatively) (BOMC-2)

সত্যি এই ধরনের কমেন্ট দেখলে মেজাজ গরম হয়।SANZEED wrote:I haven't solved it fully yet,but a useful hint:

- Sun Mar 03, 2013 12:58 pm
- Forum: International Mathematical Olympiad (IMO)
- Topic: IMO Marathon
- Replies:
**184** - Views:
**62173**

### Re: IMO Marathon

Oy Adib,are these primes $p$,$q$ distinct???????

- Sat Mar 02, 2013 3:22 pm
- Forum: Geometry
- Topic: A Very Nice Problem
- Replies:
**11** - Views:
**4806**

### Re: A Very Nice Problem

Mahi and Fahim vai, cool avatars.

This is my 200th post.

This is my 200th post.

- Thu Feb 28, 2013 1:48 pm
- Forum: Algebra
- Topic: exponential equation.
- Replies:
**4** - Views:
**2036**

### Re: exponential equation.

How could i do this kind of shitty thing?????????? I don't know who gave me the right to turn a curve into a straight line. The first mistake is where i eliminated the square. But the problem is pretty easy.