The following bound is stronger:
\[4^n=(1+1)^{2n}=\binom{2n}{0}+\cdots+\binom{2n}{2n}<(2n+1)\binom{2n}{n}\Rightarrow\binom{2n}{n}>\frac{4^n}{2n+1}\]
Search found 268 matches
- Wed Feb 18, 2015 11:06 pm
- Forum: Algebra
- Topic: power of 2 or binomial?
- Replies: 4
- Views: 6663
- Tue Feb 17, 2015 12:12 pm
- Forum: Combinatorics
- Topic: Triangle in a 2n-Graph
- Replies: 2
- Views: 2853
Re: Triangle in a 2n-Graph
This is Mantel's theorem, a special case of a more general theorem of Turan. Let us prove the following version. We prove by induction on $n$ that any graph $G$ with $n$ vertices and $>n^2/4$ edges contains a triangle. If every vertex has degree $\ge (n-1)/2$ then any two vertices must have a common...
- Sun Feb 15, 2015 9:51 pm
- Forum: Combinatorics
- Topic: triangle in a square has area less than 1/2
- Replies: 5
- Views: 8245
Re: triangle in a square has area less than 1/2
I think $(n+1)^2$ is more than sufficient, we just need $n^2+1$.
- Sat Sep 21, 2013 8:02 pm
- Forum: Secondary Level
- Topic: Help 2 prove (Combi.)
- Replies: 3
- Views: 4096
Re: Help 2 prove (Combi.)
I can think of 'A Path to Combinatorics for Undergraduates' by Titu Andreescu and Zuming Feng (not sure whether it was given to you in the camp). You can also read the chapter on combinatorics in 'Art and Craft of Problem Solving' by Paul Zeitz.
- Thu Sep 19, 2013 10:27 pm
- Forum: Secondary Level
- Topic: Help 2 prove (Combi.)
- Replies: 3
- Views: 4096
Re: Help 2 prove (Combi.)
You can use a combinatorial argument as follows: there are $a$ boys and $b$ girls in a class. We are trying to choose a team of $a$ students from among the $a+b$ students. In how many ways can this be done? The answer is clearly $\binom{a+b}{a}=\binom{a+b}{b}$. We can do the same by choosing $k$ gir...
- Sun Sep 01, 2013 6:30 pm
- Forum: News / Announcements
- Topic: Online Geometry Camp 2014 Phase 1
- Replies: 19
- Views: 15822
Re: Online Geometry Camp 2014 Phase 1
Now that the exams are over, feel free to discuss the questions.
- Sat Aug 31, 2013 9:21 pm
- Forum: National Math Camp
- Topic: [OGC1] Online Geometry Camp: Day 6 (EXAM!)
- Replies: 36
- Views: 28547
Re: [OGC1] Online Geometry Camp: Day 6 (EXAM!)
That is a reply!
- Sat Aug 31, 2013 7:45 pm
- Forum: National Math Camp
- Topic: [OGC1] Online Geometry Camp: Day 5 (EXAM!)
- Replies: 29
- Views: 24468
Re: [OGC1] Online Geometry Camp: Day 5 (EXAM!)
I think I've got all your work. (You can recheck the 'trimmed content' in my emails by clicking the button with the three dots in the bottom left corner, the trimmed bit is what I received from you.)
- Sat Aug 31, 2013 7:25 pm
- Forum: National Math Camp
- Topic: [OGC1] Online Geometry Camp: Day 5 (EXAM!)
- Replies: 29
- Views: 24468
Re: [OGC1] Online Geometry Camp: Day 5 (EXAM!)
Ira, did you email or pm me your solutions? I've checked both but can't seem to find them.
- Sat Aug 31, 2013 5:57 pm
- Forum: National Math Camp
- Topic: [OGC1] Online Geometry Camp: Day 6 (EXAM!)
- Replies: 36
- Views: 28547
Re: [OGC1] Online Geometry Camp: Day 6 (EXAM!)
I've replied to (hopefully) everyone who has sent me solutions. If you didn't get a reply, let me know asap.