## Search found 110 matches

Mon Feb 23, 2015 9:17 pm
Forum: Geometry
Topic: Italy TST 2000/2
Replies: 3
Views: 1586

i think my solution is too much boring :? let's apply Menelaus's and Menelaus's ; let , $CF\cap AB=N ; BF\cap AC=P ; BF\cap CM=R ; CF\cap BD=S ; CM\cap BD=T$ Now $\frac{BD}{DT}\frac{TF}{FA}\frac{AM}{MB}=1\Leftrightarrow \frac{AF}{FT}=\frac{BD}{DT}$ $\frac{AC}{CD}\frac{DS}{ST}\frac{TF}{FA}=1 \Leftrig... Mon Feb 23, 2015 2:47 pm Forum: Geometry Topic: USAMO 2013/1 Replies: 3 Views: 1670 ### Re: USAMO 2013/1 tanmoy wrote:You did not state what is the point M is oppsss sorry . didn't notice . now it is edited Mon Feb 23, 2015 2:38 pm Forum: Geometry Topic: Italy TST 2000/2 Replies: 3 Views: 1586 ### Italy TST 2000/2 Let$ABC$be an isosceles right triangle and$M$be the midpoint of its hypotenuse$AB$. Points$D$and$E$are taken on the legs$AC$and$BC$respectively such that$AD=2DC$and$BE=2EC$. Lines$AE$and$DM$intersect at$F$. Show that$FC$bisects the$\angle DFE$. Mon Feb 23, 2015 2:26 pm Forum: Combinatorics Topic: Circular Table conference Replies: 2 Views: 1251 ### Re: Circular Table conference if you want to know how it works , read any books of combinatorics . it could be "principles and techniques in combinatorics" or google it Mon Feb 23, 2015 2:21 pm Forum: Combinatorics Topic: Circular Table conference Replies: 2 Views: 1251 ### Re: Circular Table conference answer is$5!/2!$Sun Feb 22, 2015 11:27 pm Forum: Geometry Topic: USAMO 2013/1 Replies: 3 Views: 1670 ### Re: USAMO 2013/1 let,$M=w_{A}\cap w_{B}\cap w_{C}\frac{XY}{XZ}=\frac{sin\angle YMX \cdot YM}{sin\angle ZMX\cdot ZM}$or,$\frac{XY}{XZ}=\frac{sin\angle YMX \cdot sin\angle YZM}{sin\angle ZMX\cdot sin\angle ZYM}$now ,$\angle YMX=180-\angle MYX-\angle MXY=\angle MRB-\angle MRP=\angle PRB=\angle PMB$and let$XM\...
Thu Feb 19, 2015 12:22 am
Forum: Combinatorics
Topic: Triangle in a 2n-Graph
Replies: 2
Views: 1405

### Re: Triangle in a 2n-Graph

it is easy to check that it works for $n=2$ let it is true for $n=m$ ...... now for $n=m+1$ we have $2(m+1)=2m+2$ points and we need to connect $m^{2}+2m+2$ edges let there are $2m+2$ points in a plane such that no three are collinear .... now draw an area S such that $2m$ points are inside S. other...
Wed Feb 18, 2015 7:54 pm
Forum: Algebra
Topic: power of 2 or binomial?
Replies: 4
Views: 3137

### Re: power of 2 or binomial?

$\frac{2n}{n}=2$
$\frac{2n-1}{n-1} > 2$
$\frac{2n-2}{n-2} > 2$
.
.
$\frac{2n-(n-1)}{n-(n-1)} > 2$
multiplying all ....we have ,
$\frac{2n(2n-1)(2n-2)......(n+1)}{n(n-1)(n-2)......1}>2^{n}$
or, $\binom{2n}{n}>2^{n}$

it works for all n>1 . for n=1 both of them are equal
Sun Dec 21, 2014 2:03 am
Forum: Geometry
Topic: Side, Angle, and, (Ex)-Circle
Replies: 4
Views: 1982