## 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**

- 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
- Topic: Canada 2007
- Replies:
**1** - Views:
**935**

- Thu Jul 28, 2016 11:18 pm
- Forum: Secondary Level
- Topic: 2014-national
- Replies:
**15** - Views:
**5150**

### Re: 2014-national

This isn't obvious. You have to prove it. (This isn't even true for a $2\times 2$ chessboard.)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.

- 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)

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 $k\leq n$.jagdish wrote:Total number of whole number integer ordered pair $(n,r)$ in $\displaystyle \binom{n}{r} = 120$

- 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

Hint