## Search found 5 matches

- Tue Apr 03, 2018 11:57 pm
- Forum: Number Theory
- Topic: A Conjecture
- Replies:
**1** - Views:
**312**

### A Conjecture

Prove that for every prime $$p > 2$$,there exists a prime $$q < p$$ such that $$p-q = 2^n$$ for some $n \in N_0$.

- Mon Sep 25, 2017 3:48 am
- Forum: National Math Camp
- Topic: National Camp 2016
- Replies:
**1** - Views:
**871**

### Re: National Camp 2016

If $ABC$ is an arbitrary triangle and the value of $n$ satisfies the given condition,obviously it will be also true for certain $ABC$.Consider a triangle $ABC$ with angle $$A=120$$, $$AB=AC$$, circumcenter $O$, midpoint of $$AB=M$$, midpoint of $$AC=N$$, center of the regular polygon constructed out...

- Wed Apr 05, 2017 7:51 pm
- Forum: Number Theory
- Topic: Perfect Cube
- Replies:
**1** - Views:
**618**

### Re: Perfect Cube

Here I want to use a simple lemma,which is: Let a,b,c,d be some integer s.t. (a,b)=(c,d)=1.When $$\frac{a}{b}+\frac{c}{d}=n$$,where n \in N,then |b|=|d| It's proof is simple,first prove b|d and second d|b Proof of the original problem: First,assume (a,b,c)=1.Let (a,b)=x,(b,c)=y,(c,a)=z. $$\frac{a}{b...

- Wed Apr 05, 2017 2:29 pm
- Forum: National Math Olympiad (BdMO)
- Topic: BDMO 2017 National round Secondary 6
- Replies:
**9** - Views:
**1316**

### Re: BDMO 2017 National round Secondary 6

We can rotate the octahedron by 8\times3 or 24 ways(to prove it,just assign one side of the octahedron as A and one of it's neighbour as B and observe it's rotation). So,the total number of distinguishable octahedron is \frac{8!}{24} or 1680.

- Wed Apr 05, 2017 1:37 pm
- Forum: Combinatorics
- Topic: Binary Representation
- Replies:
**0** - Views:
**441**

### Binary Representation

Let for any positive integer $n$,$B(n)$ be the number of 1's in it's binary representation.Prove that $$B(nm) \geq \max{B(n),B(m)}$$ ,where $n,m \in N$ .