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Two circles and Three line concurrency
mofidy 0
Two circles 
and 
with equal radii intersect at P and Q. Points B and C are located on the circles and so that they are inside the circles and , respectively. Also, points X and Y distinct from P are located on and , respectively, so that:
The intersection point of the circumcircles of triangles XPC and YPB is called S. Prove that BC, XY and QS are concurrent.
Thanks.
IMAGE
and 
with equal radii intersect at P and Q. Points B and C are located on the circles and so that they are inside the circles and , respectively. Also, points X and Y distinct from P are located on and , respectively, so that:
The intersection point of the circumcircles of triangles XPC and YPB is called S. Prove that BC, XY and QS are concurrent.Thanks.
IMAGE
0 replies
A very simple question about calculus for middle school students
Craftybutterfly 11
N
by Craftybutterfly
Source: own
? (there is an easy workaround)(I know this is very easy- a little child can solve this in 1 second kinda problem so don't argue or mock me please)
11 replies
COS(Matrix)
FFA21 0
Source: My head
1)Let 
what is range of values of 
2) Let
what is range of values of
what is range of values of 
2) Let
what is range of values of
0 replies
Maximize Weighted Sum of Geometric Means
holahello 6
N
by watery
Let 
be a sequence of nonnegative real numbers whose sum is 
. Find the maximum possible value of ![$$\sum_{k=1}^\infty \left(k\cdot (3k+2) \cdot 2^{-k} \cdot \sqrt[k]{a_1a_2\dots a_k}\right).$$](//latex.artofproblemsolving.com/2/d/f/2df5179ce532077733530d5e9f1266c22d1cdb7d.png)
be a sequence of nonnegative real numbers whose sum is 
. Find the maximum possible value of ![$$\sum_{k=1}^\infty \left(k\cdot (3k+2) \cdot 2^{-k} \cdot \sqrt[k]{a_1a_2\dots a_k}\right).$$](http://latex.artofproblemsolving.com/2/d/f/2df5179ce532077733530d5e9f1266c22d1cdb7d.png)
6 replies
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
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
