Search found 62 matches

by Ragib Farhat Hasan
Thu Oct 17, 2019 3:06 am
Forum: International Mathematical Olympiad (IMO)
Topic: IMO 2019/P3
Replies: 5
Views: 57989

Re: IMO 2019/P3

Anyone looking for a classic Graph Theory problem?

Look no further!!!
by Ragib Farhat Hasan
Thu Oct 17, 2019 3:05 am
Forum: International Mathematical Olympiad (IMO)
Topic: IMO 2019/P3
Replies: 5
Views: 57989

Re: IMO 2019/P3

GRAPH THEORY... is printed all over this problem as a hint, isn't it?
by Ragib Farhat Hasan
Thu Oct 03, 2019 1:43 am
Forum: Combinatorics
Topic: Binary Representation
Replies: 1
Views: 35318

Re: Binary Representation

Can you explain the RHS of the equation $B(nm) \geq \max{B(n),B(m)}$ ?

I mean, do we multiply or, add or, individually consider the maximum values of $B(n)$ and $B(m)$?
by Ragib Farhat Hasan
Mon Sep 23, 2019 1:53 pm
Forum: National Math Olympiad (BdMO)
Topic: BdMO National Secondary 2019#6
Replies: 1
Views: 36921

Re: BdMO National Secondary 2019#6

Take one major diagonal and color the squares in black. Take the other two corners and color them in black. Take the diagonals consisting of 5 squares each (parallel to the colored major diagonal on either side) and color them in black. Therefore, total number of colored squares are $9+2+5+5=21$. It...
by Ragib Farhat Hasan
Sun Sep 15, 2019 1:49 am
Forum: Algebra
Topic: Minimum value
Replies: 4
Views: 50950

Re: Minimum value

Apparently, the denominators should be 1. So the minimum value will be 6. I don't think so. Bcoz, x>0 and y>0 are real numbers not natural. I know the difference. What I meant was: in the solution to the problem, both $x$ and $y$ should be 1. BTW, this is just an observation since I didn't go throu...
by Ragib Farhat Hasan
Mon Jan 07, 2019 9:01 pm
Forum: Social Lounge
Topic: How unfortunated we are!
Replies: 2
Views: 6949

Re: How unfortunated we are!

Of course we should. And you're right, this forum has turned into a 'death valley', which is very unfortunate. Once this forum was very active, with over 100 posts each day. But sometime in 2017 everybody got more involved with this forum called 'The Art of Problem Solving', and that was the beginni...
by Ragib Farhat Hasan
Fri Oct 19, 2018 2:04 am
Forum: National Math Olympiad (BdMO)
Topic: BdMO National Secondary 2007/12
Replies: 2
Views: 895

Re: BdMO National Secondary 2007/12

$x^2-1=0$
$x^2=1$
$x= +1$ or, $-1$

Let, $f(x)=x^{100}-2x^{51}+1$
Now, the remainder(s) can be found using the Remainder theorem.

When $x=+1$
$f(x)=1-2(1)+1$
$f(x)=0$

When $x=-1$
$f(x)=1-2(-1)+1$
$f(x)=4$
by Ragib Farhat Hasan
Fri Oct 19, 2018 1:49 am
Forum: Primary Level
Topic: Combinatorics
Replies: 6
Views: 3586

Re: Combinatorics

The answer should be 14.
by Ragib Farhat Hasan
Fri Oct 19, 2018 1:43 am
Forum: Number Theory
Topic: no solution (a,b)
Replies: 2
Views: 3733

Re: no solution (a,b)

There can be another solution to this problem, which includes a bit of messy work of Algebra. $a^2=b^7+7$ $a^2-16=b^7-9$ $(a+4)(a-4)=(\sqrt{b^7}+3)(\sqrt{b^7}-3)$ Now using a little bit of Algebra, it can be shown that there is no integer value of $a$ or, $b$ that satisfies the above equation, hence...
by Ragib Farhat Hasan
Tue Oct 16, 2018 8:01 pm
Forum: Number Theory
Topic: no solution (a,b)
Replies: 2
Views: 3733

Re: no solution (a,b)

The LHS of the equation is always positive. So $b>0$, as for $b=0, a= \sqrt7$, and when $b=-1, a=\sqrt6$. And when $b<-1$, the equation becomes invalid. Therefore, it can be deduced that $b>0$. $a^2-b^7=7$ $(a+\sqrt{b^7}) (a-\sqrt{b^7})=7$ Now, $a>b$, otherwise the equation draws into a negative res...