@Joty: I had only two to three hours to solve and latex all of them as I didn't know the date of exam. So, please inform if anything is not clear enough.
@Tiham: WLOG means Without Loss Of Generality.
Search found 153 matches
- Sat Nov 05, 2011 3:14 pm
- Forum: National Math Camp
- Topic: solutions to camp exam problem
- Replies: 28
- Views: 14574
- Sat Nov 05, 2011 11:47 am
- Forum: National Math Camp
- Topic: solutions to camp exam problem
- Replies: 28
- Views: 14574
Re: solutions to camp exam problem
I only have my PDF file. So, I'm posting it here:
- Fri Nov 04, 2011 6:59 pm
- Forum: Introductions
- Topic: Newbie From CA!!
- Replies: 5
- Views: 4904
Re: Newbie From CA!!
Welcome!
By the way, does CA mean California ?
By the way, does CA mean California ?
- Thu Nov 03, 2011 11:46 pm
- Forum: Computer Science
- Topic: gedit
- Replies: 8
- Views: 6151
Re: gedit
Your code is compiled in my pc without any problem. :-/
Try updating gcc: sudo apt-get install gcc
Try updating gcc: sudo apt-get install gcc
- Thu Nov 03, 2011 8:37 pm
- Forum: Computer Science
- Topic: gedit
- Replies: 8
- Views: 6151
Re: gedit
Go to terminal and type: sudo apt-get install gedit-plugins After installing it go to Gedit > Edit > Preference > Plugins Then tick Embedded terminal. Now write a code, suppose something.c, when you need to compile it hit Ctrl+F9 and then enter gcc something.c . If compilation is OK, enter ./a.out t...
- Tue Nov 01, 2011 10:41 pm
- Forum: National Math Olympiad (BdMO)
- Topic: Algebra: Inequalities
- Replies: 14
- Views: 8091
Re: Algebra: Inequalities
এভাবে সর্বসম্মুখে কোন কপিরাইটেড জিনিসের ডাউনলোড লিঙ্ক পোস্ট করিস না।nafistiham wrote:ifile.it/zlct48/ebooksclub.org__Inequalities__A_Mathematical_Olympiad_Approach.pdf
Use PM!
- Tue Nov 01, 2011 6:16 pm
- Forum: National Math Camp
- Topic: UK 2008: Exercise 1.97 (BOMC)
- Replies: 1
- Views: 1839
Re: UK 2008: Exercise 1.97 (BOMC)
Let, $x^2 + y^2 + z^2 = A$, $xy + yz + zx = B$
$1 = (x^3 + y^3 + z^3 - 3xyz)^2 = (x+y+z)^2 (x^2 + y^2 + z^2 -xy -yz -zx)^2$
$=(A+2B)(A-B)^2 \leq \left ( \frac{(A+2B) + (A-B) + (A-B)}{3} \right)^3 = A^3$, by A.M-G.M with equality when $A+2B=A-B$ or $B=0$
SO, minimum of $x^2 + y^2 + z^2$ is $1$.
$1 = (x^3 + y^3 + z^3 - 3xyz)^2 = (x+y+z)^2 (x^2 + y^2 + z^2 -xy -yz -zx)^2$
$=(A+2B)(A-B)^2 \leq \left ( \frac{(A+2B) + (A-B) + (A-B)}{3} \right)^3 = A^3$, by A.M-G.M with equality when $A+2B=A-B$ or $B=0$
SO, minimum of $x^2 + y^2 + z^2$ is $1$.
- Sat Oct 29, 2011 12:54 am
- Forum: International Mathematical Olympiad (IMO)
- Topic: 2003 IMO Problem 5
- Replies: 12
- Views: 8137
Re: 2003 IMO Problem 5
\[ \left( \sum^n_{i, j = 1} |x_i - x_j | \right)^2 \leq \frac{2 (n^2 - 1)}{3} \sum^n_{i, j = 1} (x_i - x_j)^2 \] \[ \Longleftrightarrow \left( \sum_{1 \leq i < j \leq n} |x_i - x_j | \right)^2 \leq \frac{(n^2 - 1)}{3} \sum_{1 \leq i < j \leq n} (x_i - x_j)^2 \] \[ \Longleftrightarrow \left( \sum_{i ...
- Mon Oct 17, 2011 2:06 pm
- Forum: Combinatorics
- Topic: China 1999
- Replies: 0
- Views: 1918
China 1999
Let, $A = \{1,2, \cdots n \}$ $\Im = \{(A_1,A_2,\cdots, A_n) | |A_i| \geq 2, |A_i \cap A_j| \leq 1 \text{ for } i\neq j \}$ And, \[ \forall \{i,j\}\subseteq A, \text{ } \forall (A_1,A_2,\cdots, A_n) \in \Im, \text{ }\exists u, 1 \leq u \leq n \text{ such that } \{i,j\} \subseteq A_u, \text{ }\{i,j\}...
- Sun Oct 16, 2011 6:53 pm
- Forum: Higher Secondary Level
- Topic: ধারার সমস্যা
- Replies: 7
- Views: 5130
Re: ধারার সমস্যা
Hint :