$\textbf{Problem} \text{ }\boxed{41}$ FInd all primes $p$ for which there exists $n\in \mathbb{N}$ so that

$$

p|n^{n+1}-(n+1)^n

$$

[Harder version: Replace $p$ with a general integer $m$]

## Search found 183 matches

- Thu Aug 11, 2016 11:51 pm
- Forum: International Mathematical Olympiad (IMO)
- Topic: IMO Marathon
- Replies:
**184** - Views:
**47419**

- Tue Aug 09, 2016 9:48 pm
- Forum: International Mathematical Olympiad (IMO)
- Topic: IMO Marathon
- Replies:
**184** - Views:
**47419**

### Re: IMO Marathon

If $a_1,a_2,a_3,a_4,a_5$ are positive reals bounded below and above by $p$ and $q$ respectively $\left(0<p\le q\right)$, then prove that \[\left(a_1+a_2+a_3+a_4+a_5\right)\left(\dfrac{1}{a_1}+\dfrac{1}{a_2}+\dfrac{1}{a_3}+\dfrac{1}{a_4}+\dfrac{1}{a_5}\right)\le 25+6\left(\sqrt{\dfrac{p}{q}}-\sqrt{\...

- Tue Aug 09, 2016 4:27 pm
- Forum: Combinatorics
- Topic: even odd even odd
- Replies:
**7** - Views:
**2362**

### Re: even odd even odd

Solution: I think my notation isn't clear. I meant when considering only values, $x_i = y_i = i$. But when I say $x_i$, I am refering to the row no. of the $i$'th row, not the column no. of the $i$'th column. Now let $\sum_{(x_i,y_j)\in S}x_i+y_j = N$ We'll show $N$ is even. First set $N = 0.$ Then...

- Tue Aug 09, 2016 10:55 am
- Forum: Combinatorics
- Topic: even odd even odd
- Replies:
**7** - Views:
**2362**

### Re: even odd even odd

I don't see how to use this hint. Write your whole solution please.Nayeemul Islam Swad wrote:Hint:

- Mon Aug 08, 2016 1:10 am
- Forum: Combinatorics
- Topic: even odd even odd
- Replies:
**7** - Views:
**2362**

### Re: even odd even odd

No, as $0$ is an even number.Golam Musabbir Joy wrote:Can any row or column be empty?

- Sun Aug 07, 2016 8:59 pm
- Forum: Combinatorics
- Topic: Maximizing edges
- Replies:
**2** - Views:
**1033**

### Re: Maximizing edges

This is just a special case of Turan's theorem.Thanic Nur Samin wrote:Let there be $n$ points in a space. Some edges are connecting them, making a graph. Maximize the number of edges so that there is no tetrahedron in the graph.

https://en.m.wikipedia.org/wiki/Tur%C3%A1n's_theorem

- Sun Aug 07, 2016 7:40 pm
- Forum: Social Lounge
- Topic: Chat thread
- Replies:
**53** - Views:
**39698**

### Re: Chat thread

I think we can revive the IMO marathon too.

- Sun Aug 07, 2016 12:17 pm
- Forum: Number Theory
- Topic: IMO Shortlist 2012 N1
- Replies:
**7** - Views:
**2161**

### Re: IMO Shortlist 2012 N1

There is no condition saying $kx^2 \in A$. You have to prove it. (Though the proof is very obvious).

- Sun Aug 07, 2016 10:19 am
- Forum: Number Theory
- Topic: IMO Shortlist 2012 N1
- Replies:
**7** - Views:
**2161**

### Re: IMO Shortlist 2012 N1

OK. I have approached it like this. Let, $d= gcd(m,n).$ If, $d > 1$, if $d|x$ & $ d|y $ then $d|x^2+kxy+y^2 $ for all k.So,though the multiple of d satisfies condition, but $A \ne \mathbb{Z}$. here it is easy to see, $ d=1$ . Now,We let,the set $A$ is admissible containing m,n.So,if $ x^2 \in A$ , ...

- Fri Aug 05, 2016 8:26 pm
- Forum: Combinatorics
- Topic: Guide the rook
- Replies:
**1** - Views:
**998**

### Re: Guide the rook

Hint: