## BdMO National Higher Secondary :Problem Collection(2016)

samiul_samin
Posts: 1007
Joined: Sat Dec 09, 2017 1:32 pm

### BdMO National Higher Secondary :Problem Collection(2016)

You will get question $1,2,3,4,5,6,7,8$ herehttp://matholympiad.org.bd/forum/viewto ... 878#p17476

Now the number 9 and The Final Question

9.
The integral $Z(0)=\int^{\infty}_{-\infty} dx e^{-x^2}= \sqrt{\pi}$

(a)(3 POINTS:)Show that the integral $Z(j)=\int^{\infty}_{-\infty} dx e^{-x^{2}+jx}$
Where $j$ is not a function of $x$,is $Z(j)=e^{j^{2}/4a} Z(0)$

(b)(10 POINTS):Show that,
$\dfrac 1 {Z(0)}=\int x^{2n} e^{-x^2}= \dfrac {(2n-1)!!}{2^n}$
Where $(2n-1)!!$ is defined as $(2n-1)(2n-3)\times...\times3\times 1$

(c)(7 POINTS):What is the number of ways to form $n$ pairs from $2n$ distinct objects?Interept the previous part of the problem in term of this answer.

[It was a 200 number exam,and this is one of the toughest problems.]

samiul_samin
Posts: 1007
Joined: Sat Dec 09, 2017 1:32 pm

### Re: BdMO National Higher Secondary :Problem Collection(2016)

Counting in this way and applying Multiplication Principle we get the number of ways to make $n$ pairs from $2n$ object is $(2n-3)×(2n-1)...3×1$