Prove that, for $a+b+c+d+e=3$,
\[ \left( \frac{a}{1-a} \right) \left( \frac{b}{1-b} \right) \left( \frac{c}{1-c} \right) \left( \frac{d}{1-d} \right) \left( \frac{e}{1-e} \right) \ge \left( \frac{3}{2} \right)^{5} \]
Search found 153 matches
- Tue Dec 21, 2010 11:09 pm
- Forum: Algebra
- Topic: a + b + c + d + e = 3
- Replies: 1
- Views: 2024
- Tue Dec 21, 2010 11:02 pm
- Forum: Algebra
- Topic: a + b + c = 1
- Replies: 0
- Views: 4714
a + b + c = 1
Prove that, for $a+b+c=1$,
\[ \sum_{a,b,c}{\left(\frac{2}{a^{2}}+\frac{7}{a}\right)}\ge 9+\frac{4}{abc} \]
\[ \sum_{a,b,c}{\left(\frac{2}{a^{2}}+\frac{7}{a}\right)}\ge 9+\frac{4}{abc} \]
- Tue Dec 21, 2010 11:00 pm
- Forum: Algebra
- Topic: n variable nice inequality
- Replies: 7
- Views: 4317
n variable nice inequality
Prove that, for $a_1, a_2, \cdots a_n \geq 0$,
\[ \left( a_{1}+\frac{a_{2}}{2}+\cdots+\frac{a_{n}}{n}\right) (a_{1}+2 a_{2}+\cdots+n a_{n})\leq \frac{(n+1)^{2}}{4n} (a_{1}+a_{2}+\cdots+a_{n})^{2} \]
\[ \left( a_{1}+\frac{a_{2}}{2}+\cdots+\frac{a_{n}}{n}\right) (a_{1}+2 a_{2}+\cdots+n a_{n})\leq \frac{(n+1)^{2}}{4n} (a_{1}+a_{2}+\cdots+a_{n})^{2} \]
- Tue Dec 21, 2010 11:00 pm
- Forum: Algebra
- Topic: A 3 variable inequality
- Replies: 2
- Views: 2200
A 3 variable inequality
prove that, for positive reals $a, b, c$,
\[ \frac{a^{2}}{bc}+\frac{b^{2}}{ac}+\frac{c^{2}}{ab}\ge \frac{a^{2}+b^{2}+c^{2}} {ab+bc+ca} + \frac{4}{3} \left( \frac{ab}{a^{2}+b^{2}} + \frac{bc}{b^{2}+c^{2}}+\frac{ca}{c^{2}+a^{2}} \right) \]
\[ \frac{a^{2}}{bc}+\frac{b^{2}}{ac}+\frac{c^{2}}{ab}\ge \frac{a^{2}+b^{2}+c^{2}} {ab+bc+ca} + \frac{4}{3} \left( \frac{ab}{a^{2}+b^{2}} + \frac{bc}{b^{2}+c^{2}}+\frac{ca}{c^{2}+a^{2}} \right) \]
- Tue Dec 21, 2010 10:56 pm
- Forum: Algebra
- Topic: N'th Differencial
- Replies: 3
- Views: 2938
N'th Differencial
Prove that, for any function $f(x)$,
\[ \frac{{d^{n}}}{{dx^{n}}}f(x)=\lim_{h\to 0}\frac{\sum_{i=0}^{n}{(-1)^{i}}\binom{n}{i}f(x+(n-i)h)}{h^{n}} \]
\[ \frac{{d^{n}}}{{dx^{n}}}f(x)=\lim_{h\to 0}\frac{\sum_{i=0}^{n}{(-1)^{i}}\binom{n}{i}f(x+(n-i)h)}{h^{n}} \]
- Tue Dec 21, 2010 9:11 pm
- Forum: Algebra
- Topic: Some Fibonacchi!!
- Replies: 14
- Views: 9428
Some Fibonacchi!!
Fibonacchi series is defined by, i) $F_0 = F_1 = 1$ ii) $F_{i+2} = F_{i+1}+F_i$ for all $i\geq 0 $ Now, Prove that, \[1. \sum_{i\geq 0}{\frac{F_i}{F_{i+1}F_{i+2}}} = 1\] \[2. \sum_{i\geq 0}{\frac{1}{F_i}}\text{ converges [ from Nayel vai ]}\] \[3. \sum_{i\geq 0}{\frac{(i+1)F_i}{F_{i+1}F_{i+2}}}\text...
- Mon Dec 20, 2010 10:03 pm
- Forum: Social Lounge
- Topic: Favorite mathematician?
- Replies: 35
- Views: 76988
Re: Favorite mathematician?
লিওনার্দ অয়লার
- Mon Dec 20, 2010 9:56 pm
- Forum: International Olympiad in Informatics (IOI)
- Topic: problem!problem!
- Replies: 2
- Views: 3200
Re: problem!problem!
Let the n numbers stored in an array a[n], then the algo will be:
max = a[0]
for ( i = 1 ; i < n ; i++)
max = ( a >= 0 ) ? ( a + max ) : ( ( a >= max ) ? a : max ) )
max = a[0]
for ( i = 1 ; i < n ; i++)
max = ( a >= 0 ) ? ( a + max ) : ( ( a >= max ) ? a : max ) )
- Sat Dec 18, 2010 3:15 am
- Forum: Geometry
- Topic: New problem
- Replies: 2
- Views: 3436
Re: New problem
HINT:
- Thu Dec 09, 2010 6:36 pm
- Forum: News / Announcements
- Topic: ১০০তম পোস্ট
- Replies: 4
- Views: 4515
Re: ১০০তম পোস্ট
প্রথম লাইন পড়ে খাট থেকে পড়ে গেসিলাম। উঠে দ্বিতীয় লাইন পড়ে শান্তি লাগতেসে