Prove that, all the numbers in the $2^k-th$ row of the Pascal's Triangle, except the end one's, are even, i.e,
\[\displaystyle2\mid\binom{2^k}{r}\]
\[\displaystyle\forall 0<r<2^k\]
Search found 155 matches
- Thu Aug 16, 2012 10:38 pm
- Forum: Combinatorics
- Topic: Factors of Binomials:Part 2
- Replies: 5
- Views: 3977
- Thu Aug 16, 2012 4:30 pm
- Forum: Combinatorics
- Topic: Factors of Binomials
- Replies: 4
- Views: 3902
Re: Factors of Binomials
Lemma:
- Thu Aug 16, 2012 4:27 pm
- Forum: Combinatorics
- Topic: Factors of Binomials
- Replies: 4
- Views: 3902
Factors of Binomials
Suppose, $p$ is a prime. Show that, all the numbers in the $p^{th}$ row of the Pascal's Triangle (except $\binom{p}{0}$ and $\binom{p}{p}$) are divisible by $p$, i.e,
\[\displaystyle p\mid \binom{p}{k}\]
\[\displaystyle\forall 0<k<p\]
\[\displaystyle p\mid \binom{p}{k}\]
\[\displaystyle\forall 0<k<p\]
- Thu Aug 16, 2012 2:54 pm
- Forum: Combinatorics
- Topic: A Multinomial Conundrum
- Replies: 0
- Views: 1745
A Multinomial Conundrum
A polynomial in $x$ is defined by
$$a_0+a_1x+a_2x^2+...+a_{2n}x^{2n}=(x+2x^2+3x^3+...+nx^n)^2$$
Show that,
\[\displaystyle\sum_{i=n+1}^{2n}a_i=\frac{n(n+1)(5n^2+5n+2)}{24}\]
$$a_0+a_1x+a_2x^2+...+a_{2n}x^{2n}=(x+2x^2+3x^3+...+nx^n)^2$$
Show that,
\[\displaystyle\sum_{i=n+1}^{2n}a_i=\frac{n(n+1)(5n^2+5n+2)}{24}\]
- Thu Aug 09, 2012 5:49 pm
- Forum: Asian Pacific Math Olympiad (APMO)
- Topic: APMO 1990 (Inequality with Combi)
- Replies: 2
- Views: 3598
Re: APMO 1990 (Inequality with Combi)
Vaia, good use of the AM-GM inequality. Thanks for the post.
- Wed Aug 08, 2012 8:40 pm
- Forum: Asian Pacific Math Olympiad (APMO)
- Topic: APMO 1990 (Inequality with Combi)
- Replies: 2
- Views: 3598
APMO 1990 (Inequality with Combi)
Let, $a_1,a_2,...,a_n$ be positive real numbers, and let $S_k$ be the sum of the products of $a_1,a_2,...,a_n$ taken $k$ at a time. Show that,
\[\displaystyle S_kS_{n-k}\geq \binom{n}{k}^2a_1a_2...a_n\]
\[\forall k=1,2,...,n-1\]
\[\displaystyle S_kS_{n-k}\geq \binom{n}{k}^2a_1a_2...a_n\]
\[\forall k=1,2,...,n-1\]
- Mon Jul 16, 2012 11:22 pm
- Forum: News / Announcements
- Topic: IMO-2012 result of Bangladesh team
- Replies: 4
- Views: 6899
Re: IMO-2012 result of Bangladesh team
Well, it's official....!Bappa Vai has got the 1st silver on behalf of Bangladesh, Sourav Vai and Mahi have both got Bronze medals, and Mugdho Vai and Adib have got HMs. Again, what a result!!! They are truly worth being called heroes. Congrats.
- Sun Jul 15, 2012 7:21 pm
- Forum: News / Announcements
- Topic: IMO-2012 result of Bangladesh team
- Replies: 4
- Views: 6899
Re: IMO-2012 result of Bangladesh team
কোপাআআআ......শামসু। (Sorry for the language. But, I am just really excited. )
What a brilliant performance !!! Hats off to all of them for giving their all and making Bangladesh proud.
Claps and Cheers. Best of luck to all of them.
What a brilliant performance !!! Hats off to all of them for giving their all and making Bangladesh proud.
Claps and Cheers. Best of luck to all of them.
- Thu Jul 05, 2012 8:22 pm
- Forum: Physics
- Topic: Congrats, CERN :)
- Replies: 2
- Views: 3068
Re: Congrats, CERN :)
Yes, Bhaia. They are not completely sure if it's Higg's Boson or not. But, many of its characters have matched with the Higg's Boson. So, their assumption is more on the side of it being the Higg's Boson. I should have stated that in the post. Sorry.... :oops: Nevertheless, it is still quite a remar...
- Thu Jul 05, 2012 6:59 pm
- Forum: Junior Level
- Topic: Final two numbers
- Replies: 1
- Views: 2201
Re: Final two numbers
Hint:
Solution: