## Search found 19 matches

- Wed May 15, 2019 2:34 pm
- Forum: Number Theory
- Topic: Difference Between Divisors
- Replies:
**1** - Views:
**246**

### Re: Difference Between Divisors

Solution: We will prove that there are infinitely many positive integers $n$ and $a$ such that $n^2+1=a(a-n)$. LEMMA: There are infinitely many positive integer solutions to the equation $5x^2-4=y^2$. Proof: Assume that $(x',y')$ is a solution of this equation.Then we follow the transformation: $(x,...

- Wed May 15, 2019 2:12 pm
- Forum: Number Theory
- Topic: Difference Between Divisors
- Replies:
**1** - Views:
**246**

### Difference Between Divisors

Show that there are infinitely many positive integers $n$ such that $n^2+1$ has two positive integer divisors whose difference is $n$.

- Thu Mar 28, 2019 9:14 pm
- Forum: National Math Olympiad (BdMO)
- Topic: BdMO National Higher Secondary 2019/9
- Replies:
**2** - Views:
**422**

### Re: BdMO National Higher Secondary 2019/9

Let $B'$ be a point on the line $AB$ such that $AB'=AC$ and $C'$ be a point on the line $DC$ such that $DC'=BD$. So, it suffices to prove that $BB'=CC'$. $\triangle AB'P \cong \triangle ACP$ $ \Rightarrow \angle APB'=\angle APC$ $ \Rightarrow \angle BPB'=\angle APD$. Similiarly, $\triangle DBP \cong...

- Tue Mar 19, 2019 4:57 pm
- Forum: National Math Olympiad (BdMO)
- Topic: BdMO National Higher Secondary 2019/5
- Replies:
**1** - Views:
**387**

### Re: BdMO National Higher Secondary 2019/5

We will prove it by strong Induction. Call a permutation $a_1,a_2,....,a_n$ "GOOD" if the average of any two numbers doesn't appear in between them and call it "VERY GOOD" if it is "GOOD" and for every integer $x\in {1,2,...,n} $, there exists a $i$ such that $a_i=x$ Lemma: If $a_1,a_2,....,a_n$ is ...

- Sat Jan 19, 2019 9:17 pm
- Forum: Combinatorics
- Topic: ISL 2010 C2
- Replies:
**1** - Views:
**272**

### Re: ISL 2010 C2

Solution: We claim that the smallest positive integer $M$ is $2^{N-2}+1$. We will prove that the highest positive integer $K$ such that we will not find a $DIVERSE$ set of $K$ flags is $2^{N-2}$. A set of $K$ flags is not $DIVERSE$ if and only if there exist two positive integers $m$ and $n$ such th...

- Sat Jan 19, 2019 8:46 pm
- Forum: Combinatorics
- Topic: ISL 2010 C2
- Replies:
**1** - Views:
**272**

### ISL 2010 C2

On some planet, there are $2^N$ countries $(N\geq 4)$.Each country has a flag $N$ units wide and $1$ unit high composed of $N$ fields of size $1\times1$, each field being either $Yellow$ or $Blue$.No two countries have the same flag.We say that a set of $N$ flags is $DIVERSE$ if these flags can be a...

- Sat Apr 14, 2018 12:43 pm
- Forum: Combinatorics
- Topic: Football and Combi
- Replies:
**3** - Views:
**557**

### Re: Football and Combi

why?samiul_samin wrote: ↑Fri Apr 13, 2018 9:54 pmIt is not possible if any of the referees is not same weighted.

- Thu Apr 12, 2018 2:32 pm
- Forum: Combinatorics
- Topic: Football and Combi
- Replies:
**3** - Views:
**557**

### Football and Combi

$23$ people, each with integral weight,decide to play football, separating into two teams of $11$ people and a referee.To keep things fair, the teams chosen must have the equal total weight.It turns out that no matter who is chosen to be the referee, this can always be done.Prove that the $23$ peopl...

- Tue Mar 13, 2018 6:36 pm
- Forum: National Math Olympiad (BdMO)
- Topic: BdMO National Secondary/Higher Secondary 2018/6
- Replies:
**3** - Views:
**736**

### Re: BdMO National Higher Secondary 2018#6

$3(m^2+n^2)- 7(m+n) =-4$ Substitute $(m+n)=x$ and $(m^2+n^2)=y$ So, $7x-3y=4$ and the general solution of this diophantine equation is $x=3k+1$, $y=7k+1$ where $k$ is an integer. So, $(m+n)=3k+1$............$(1)$ $(m^2+n^2)=7k+1$...........$(2)$ From $(2)$ we get that $m^2+n^2=7k+1$ or,$[(m+n)^2 +(m...

- Tue Mar 13, 2018 5:33 pm
- Forum: National Math Olympiad (BdMO)
- Topic: BdMO National Secondary/Higher Secondary 2018/3
- Replies:
**2** - Views:
**563**

### Re: BdMO National Higher Secondary 2018#3

The ans. is $18$