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Distribution of prime numbers
Rainbow1971 3
N
by Rainbow1971
Could anybody possibly prove that the limit of 
is 
, maybe even with rather elementary means? As usual, 
denotes the 
-th prime number. The problem of that limit came up in my partial solution of this problem: https://artofproblemsolving.com/community/c7h3495516.
Thank you for your efforts.
is 
, maybe even with rather elementary means? As usual, 
denotes the 
-th prime number. The problem of that limit came up in my partial solution of this problem: https://artofproblemsolving.com/community/c7h3495516.Thank you for your efforts.
3 replies
limsup a_n/n^4
EthanWYX2009 3
N
by loup blanc
Source: 2023 Aug taca-15
Let 
. Define 
as the average of all 
for 
. Determine the value of ![\[ a = \lim_{k \to \infty} \sup_{n \geq k} \frac{a_n}{n^4} .\]](//latex.artofproblemsolving.com/5/2/d/52d93af7aafa6657689246d9a74c8eb05d96efa3.png)
. Define 
as the average of all 
for 
. Determine the value of ![\[ a = \lim_{k \to \infty} \sup_{n \geq k} \frac{a_n}{n^4} .\]](http://latex.artofproblemsolving.com/5/2/d/52d93af7aafa6657689246d9a74c8eb05d96efa3.png)
3 replies
Chebyshev polynomial and prime number
mofidy 2
N
by mofidy
Let 
be a Chebyshev polynomial of the second kind. If n>2 and x > 2 is a integer, Could 
be a prime number?
Thanks.
be a Chebyshev polynomial of the second kind. If n>2 and x > 2 is a integer, Could 
be a prime number?Thanks.
2 replies
Romanian National Olympiad 2024 - Grade 11 - Problem 1
Filipjack 4
N
by Fibonacci_math
Source: Romanian National Olympiad 2024 - Grade 11 - Problem 1
Let 
be an open interval and 
a twice differentiable function such that 
for any 
Prove that 
for any
be an open interval and 
a twice differentiable function such that 
for any 
Prove that 
for any
4 replies
Romania NMO 2023 Grade 11 P1
DanDumitrescu 14
N
by Rohit-2006
Source: Romania National Olympiad 2023
Determine twice differentiable functions 
which verify relation
![\[
\left( f'(x) \right)^2 + f''(x) \leq 0, \forall x \in \mathbb{R}.
\]](//latex.artofproblemsolving.com/3/a/2/3a235e8be4c61c32f376999e61d64973db21dc75.png)
which verify relation![\[
\left( f'(x) \right)^2 + f''(x) \leq 0, \forall x \in \mathbb{R}.
\]](http://latex.artofproblemsolving.com/3/a/2/3a235e8be4c61c32f376999e61d64973db21dc75.png)
14 replies
f(x)<=f(a) for all a and all x in a left neighbour of a implies monotony if cont
CatalinBordea 7
N
by solyaris
Source: Romanian District Olympiad 2012, Grade XI, Problem 4
A function 
has property 
if for any real number 
there exists a 
such that 
for all 
a) Give an example of a function with property
that is not monotone on 
b) Prove that a continuous function that has property is nondecreasing.
has property 
if for any real number 
there exists a 
such that 
for all 
a) Give an example of a function with property
that is not monotone on 
b) Prove that a continuous function that has property
is nondecreasing.
7 replies
vectorspace
We2592 1
N
by Acridian9
Q.) Let 
where
is field. Define addition of elements of 
coordinate wise and for 
and 
define 
.
Is is a vector space over field
how to solve it please help
where
is field. Define addition of elements of 
coordinate wise and for 
and 
define 
.Is
is a vector space over field how to solve it please help
1 reply
Rigid sets of points
a_507_bc 3
N
by solyaris
Source: ICMC 8.1 P6
A set of points in the plane is called rigid if each point is equidistant from the three (or more) points nearest to it.
(a) Does there exist a rigid set of
points?
(b) Does there exist a rigid set of
points?
(a) Does there exist a rigid set of
points?(b) Does there exist a rigid set of
points?
3 replies
Finding pairs of functions of class C^2 with a certain property
Ciobi_ 3
N
by solyaris
Source: Romania NMO 2025 11.1
Find all pairs of twice differentiable functions 
, with their second derivative being continuous, such that the following holds for all 
: ![\[(f(x)-g(y))(f'(x)-g'(y))(f''(x)-g''(y))=0\]](//latex.artofproblemsolving.com/0/2/9/029a935e28ca0a9b36e65702689cca91eeb8f614.png)
, with their second derivative being continuous, such that the following holds for all 
: ![\[(f(x)-g(y))(f'(x)-g'(y))(f''(x)-g''(y))=0\]](http://latex.artofproblemsolving.com/0/2/9/029a935e28ca0a9b36e65702689cca91eeb8f614.png)
3 replies
