Re: BdMO National Secondary: Problem Collection(2016)
Posted: Mon Feb 05, 2018 10:39 pm
by Tasnood
5(b)
We can set the Earthlings in $n!$ ways if they are in straight line. But considering rotation, there exists a group of $n$ arrangements, that are actually same if we rotate them.
If $n=3$, rearrangements are: $ABC,BCA,CAB,ACB,CBA,BAC$, where the first three are same and so are the last three.
So, actual arrangements=$2$=$\frac{3!}{3}$. We can say:$\frac{n!}{n}$ [Can check for $n=4$]
The rest is same. $m$ Martians can be placed in $n$ places in $n \choose m$ ways. And, the Martians can be arranged in $m!$ ways.
So, the result=$\frac{n!}{n}.m!.{n \choose m}$=$\frac{n.{(n-1)!}^2}{(n-m)!}$
Re: BdMO National Secondary: Problem Collection(2016)
Posted: Tue Feb 13, 2018 8:37 am
by samiul_samin
Where is the problem 7?
Re: BdMO National Secondary: Problem Collection(2016)
Re: BdMO National Secondary: Problem Collection(2016)
Posted: Mon Feb 18, 2019 1:35 pm
by prottoy das
Let $CH\perp AB$. $BI$ & $DO$ meet at $E$. Now $BIOD$ cyclic so, $\angle IBD=\angle EOI$. Again, as $\angle ABC$ is bisected by $BI$ so $\angle IBD= \angle IBH$. So,$\angle IBH=\angle EBH=\angle EOH$, so the quadrangle $EDBH$ is cyclic. So $\angle BEO=\angle BHO=90^\circ$
Re: BdMO National Secondary: Problem Collection(2016)
Let $CH\perp AB$. $BI$ & $DO$ meet at $E$. Now $BIOD$ cyclic so, $\angle IBD=\angle EOI$. Again, as $\angle ABC$ is bisected by $BI$ so $\angle IBD= \angle IBH$. So,$\angle IBH=\angle EBH=\angle EOH$, so the quadrangle $EDBH$ is cyclic. So $\angle BEO=\angle BHO=90^\circ$
Problem no $6(b)$.
Mension the problem number to easily understand the solution.
Re: BdMO National Secondary: Problem Collection(2016)
Posted: Mon Feb 18, 2019 5:14 pm
by samiul_samin
Problem 1(a)
Hard solution
$(n-1)×n×(n+1)=(n^2-n)(n+1)$[According to fermat's little theorem given expression is divided by $2$ ]
$(n-1)×n×(n+1)=(n^2-n)(n+1)=n^3-n$[According to fermat's little theorem given expression is divided by $3$ ]
So,$(n-1)×n×(n+1)$ is divided by $2×3$ or $6$. Proved.