Search found 19 matches

by Abdullah Al Tanzim
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,...
by Abdullah Al Tanzim
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 $n$.
by Abdullah Al Tanzim
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...
by Abdullah Al Tanzim
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 ...
by Abdullah Al Tanzim
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...
by Abdullah Al Tanzim
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...
by Abdullah Al Tanzim
Sat Apr 14, 2018 12:43 pm
Forum: Combinatorics
Topic: Football and Combi
Replies: 3
Views: 557

Re: Football and Combi

samiul_samin wrote:
Fri Apr 13, 2018 9:54 pm
It is not possible if any of the referees is not same weighted.
why?
by Abdullah Al Tanzim
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...
by Abdullah Al Tanzim
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...