Search found 183 matches

by asif e elahi
Fri Aug 05, 2016 7:53 pm
Forum: Asian Pacific Math Olympiad (APMO)
Topic: APMO 2016 #4
Replies: 2
Views: 5382

Re: APMO 2016 #4

I think, the answer is $57$.(may be). According to condition, we can make a cycle of $57$ cities where it is always possible to have a reach-connection between any two cities using at most $28$ flights . So,any $2$ of those $57$ cities cannot situate in the same group. So, at least $57$ groups are ...
by asif e elahi
Fri Aug 05, 2016 2:07 pm
Forum: Geometry
Topic: Circumcircle is tangent to the circumcircle
Replies: 2
Views: 1098

Re: Circumcircle is tangent to the circumcircle

Step 1: Let $\overrightarrow{MH}\cap \bigodot ABC=U$. Prove $U\in \bigodot ABC$.
Step 2: Let $\overrightarrow{AH}\cap \bigodot ABC=T$. Prove $T\in US$.
by asif e elahi
Fri Aug 05, 2016 12:35 pm
Forum: Site Support
Topic: Visual problems with align code
Replies: 3
Views: 7800

Re: Visual problems with align code

Well, I don't see any overlapping. It's just fine.
by asif e elahi
Wed Aug 03, 2016 11:36 am
Forum: Combinatorics
Topic: Canada 2007
Replies: 1
Views: 935

Re: Canada 2007

Divide the board into two equal parts by drawing a zigzagged diagonal and color each of them like a chessboard.
by asif e elahi
Thu Jul 28, 2016 11:18 pm
Forum: Secondary Level
Topic: 2014-national
Replies: 15
Views: 5150

Re: 2014-national

RJHridi wrote: So the maximum number of knights placed in the board will be the number of black or white squares, assuring that none attacks anyone.
This isn't obvious. You have to prove it. (This isn't even true for a $2\times 2$ chessboard.)
Try to find a mapping from the white squares to the black squares.
by asif e elahi
Thu Jul 28, 2016 10:26 pm
Forum: Geometry
Topic: Incenter of Triangle
Replies: 2
Views: 1346

Re: Incenter of Triangle

Take the triange $ABC$ for which it attains the minimum value.Consider the case when $CA\neq CB$. Now let $CI$ meets the incirle at $D$ so that $I$ lies between $C$ and $D$. Let the tangent at $D$ meet $AI,BI$ at $A_0,B_0$ resp. Then prove that $AI+BI \geq A_0+B_0$(You will need to go through some m...
by asif e elahi
Thu Jul 28, 2016 9:56 pm
Forum: Higher Secondary Level
Topic: ordered pair (n,r)
Replies: 1
Views: 1141

Re: ordered pair (n,r)

jagdish wrote:Total number of whole number integer ordered pair $(n,r)$ in $\displaystyle \binom{n}{r} = 120$
You can use the well known fact that $\frac{n}{gcd(n,k)} \mid \binom{n}{k} \text{ } \forall n,k\in \mathbb{N}$ with $k\leq n$.
by asif e elahi
Tue May 31, 2016 5:04 pm
Forum: Combinatorics
Topic: even odd even odd
Replies: 7
Views: 2363

even odd even odd

A $2015\times 2015$ grid is coloured like a chessboard so that the four corner squres are coloured black. We put pebbles in some of the cells so that every row and column contains an odd number of cells with pebbles. Prove that there are an even number of white cells with pebbles.
by asif e elahi
Tue May 31, 2016 4:53 pm
Forum: Algebra
Topic: Inequality with xyz=1
Replies: 1
Views: 1291

Re: Inequality with xyz=1

Let \begin{align*} \dfrac{1}{2x-1}=a\\ \dfrac{1}{2y-1}=b\\ \dfrac{1}{2z-1}=c \end{align*} So \begin{align*} x=\dfrac{a+1}{2a}\\ y=\dfrac{b+1}{2b}\\ z=\dfrac{c+1}{2c} \end{align*} Now $xyz=1$ implies $(a+1)(b+1)(c+1)=8abc$ So \begin{align*} 8abc & = \prod\limits_{cyc}^{}(a+1)\\ & \geq \prod\limits_{c...
by asif e elahi
Wed Apr 27, 2016 4:17 pm
Forum: Geometry
Topic: feet of perpendicular and midpoint
Replies: 1
Views: 1076

Re: feet of perpendicular and midpoint

It will be equal to $0$ if you take directed segment, otherwise not.

Hint
Write $\vec{LP}.\vec{BC}$ in terms of $\vec{BP}$ and$\vec{CP}$.