Search found 22 matches

by Abdullah Al Tanzim
Sun Oct 20, 2019 11:04 pm
Forum: International Mathematical Olympiad (IMO)
Topic: IMO '94 P3
Replies: 1
Views: 212

Re: IMO '94 P3

Solution: (a) Let $A= a_1,a_2,a_3,.... $ be the sequence of positive integers whose base $2$ representation has precisely three $1s$ and for all $k\in N$, $a_{k+1}>a_k$. Claim 1 : $0\leq f(k+1)-f(k)\leq 1$ Proof: Note that $2k+2 \in A$ iff $k+1 \in A$. Now, $f(k+1)-f(k)<0$ implies $k+1$ is in $A$ an...
by Abdullah Al Tanzim
Sun Oct 20, 2019 8:58 pm
Forum: International Mathematical Olympiad (IMO)
Topic: IMO '94 P3
Replies: 1
Views: 212

IMO '94 P3

For any positive integer $k$, let $f(k)$ be the number of elements in the set ${k+1,k+2,...,2k}$ whose base $2$ representation has precisely three $1s$. (a) Prove that for each positive integer $m$, there exists at least one positive integer $k$ such that $f(k)=m$. (b) Determine all positive integer...
by Abdullah Al Tanzim
Tue Jul 23, 2019 12:31 am
Forum: International Mathematical Olympiad (IMO)
Topic: IMO 2019/P4
Replies: 1
Views: 513

Re: IMO 2019/P4

Solution: We will show that $(1,1)$ and $(3,2)$ are the only possible pairs. Claim1: $\lfloor\frac{k}{3}\rfloor =\lfloor\frac{n}{2}\rfloor$ Proof: $k!=(2^n-1)(2^n-2)(2^n-4)......(2^n-2^{n-1})$ $ =2^{\frac{n(n-1)}{2}}(2^n-1)(2^{n-1}-1)(2^{n-2}-1)....(2-1)$ ...............(1) As $2$ is a primitive roo...
by Abdullah Al Tanzim
Wed May 15, 2019 2:34 pm
Forum: Number Theory
Topic: Difference Between Divisors
Replies: 1
Views: 579

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: 579

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: 734

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: 640

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: 453

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: 453

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: 828

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?