A complex integral
by Gauler, Mar 30, 2025, 3:26 PM
For 
with 
, prove that
for 
.
with 
, prove that
for 
.
something like MVT
by mqoi_KOLA, Mar 29, 2025, 11:37 AM
If 
is a continuous function on ![$[0,1]$](//latex.artofproblemsolving.com/e/8/6/e861e10e1c19918756b9c8b7717684593c63aeb8.png)
such that 
, then there exists a 
such that:
![\[
F(c) = \frac{1}{c} \int_0^c F(x) \,dx
\]](//latex.artofproblemsolving.com/8/f/e/8fea4d84bc98b6f4953e19976cfa554945a37ea6.png)
is a continuous function on ![$[0,1]$](http://latex.artofproblemsolving.com/e/8/6/e861e10e1c19918756b9c8b7717684593c63aeb8.png)
such that 
, then there exists a 
such that:![\[
F(c) = \frac{1}{c} \int_0^c F(x) \,dx
\]](http://latex.artofproblemsolving.com/8/f/e/8fea4d84bc98b6f4953e19976cfa554945a37ea6.png)
This post has been edited 2 times. Last edited by mqoi_KOLA, Yesterday at 11:39 AM
Gheorghe Țițeica 2025 Grade 12 P4
by AndreiVila, Mar 28, 2025, 10:05 PM
Let 
be a ring. Let 
such that 
. Prove that if 
is nilpotent, so is 
.
Janez Šter
be a ring. Let 
such that 
. Prove that if 
is nilpotent, so is 
.Janez Šter
Gheorghe Țițeica 2025 Grade 12 P2
by AndreiVila, Mar 28, 2025, 10:01 PM
Let ![$f:[0,1]\rightarrow\mathbb{R}$](//latex.artofproblemsolving.com/9/d/0/9d00bb0f11e69022dded4a8baba1b496f66d272e.png)
be a continuous function. Prove that 
![$f:[0,1]\rightarrow\mathbb{R}$](http://latex.artofproblemsolving.com/9/d/0/9d00bb0f11e69022dded4a8baba1b496f66d272e.png)
be a continuous function. Prove that 
Gheorghe Țițeica 2025 Grade 11 P2
by AndreiVila, Mar 28, 2025, 9:42 PM
Let 
and 
such that 
Prove that 
for all 
.
Cristi Săvescu
and 
such that 
Prove that 
for all 
.Cristi Săvescu
Gheorghe Țițeica 2025 Grade 11 P1
by AndreiVila, Mar 28, 2025, 9:40 PM
Find all continuous functions 
such that 
for all 
.
such that 
for all 
.
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