## Search found 183 matches

Thu Aug 11, 2016 11:51 pm
Topic: IMO Marathon
Replies: 184
Views: 47419

### Re: IMO Marathon

$\textbf{Problem} \text{ }\boxed{41}$ 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 $m$]
Tue Aug 09, 2016 9:48 pm
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.
Mon Aug 08, 2016 1:10 am
Forum: Combinatorics
Topic: even odd even odd
Replies: 7
Views: 2362

### Re: even odd even odd

Golam Musabbir Joy wrote:Can any row or column be empty?
No, as $0$ is an even number.
Sun Aug 07, 2016 8:59 pm
Forum: Combinatorics
Topic: Maximizing edges
Replies: 2
Views: 1033

### Re: Maximizing edges

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.
This is just a special case of Turan's theorem.
https://en.m.wikipedia.org/wiki/Tur%C3%A1n's_theorem
Sun Aug 07, 2016 7:40 pm
Forum: Social Lounge
Replies: 53
Views: 39698

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

Hint: