## Search found 183 matches

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 ...
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

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.
Wed Aug 03, 2016 11:36 am
Forum: Combinatorics
Replies: 1
Views: 935

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.)
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...
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 $k\leq n$.
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.
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...
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