AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
of positive integers is called central if for every positive integer 
, the arithmetic mean of the first 
terms of the sequence is equal to .
, 
, 
, 
of positive integers such that for every central sequence 
, 
, 
, , there are infinitely many positive integers with 
.
, let 
be all the positive integers smaller than that are coprime to . Find all 
such that
for all 
.
be positive real numbers such that : 
. Prove that : 
satisfies the equation
, find all possible positive integer values of![\[
\frac{a}{b} + \frac{b}{c} + \frac{c}{a}.
\]](http://latex.artofproblemsolving.com/3/2/b/32b41635c159921e11245369e66185ec69b0d785.png)
with the following property: the sum of the elements of any finite subset of is not a power of an integer with an exponent greater than 1?
be the n-th prime number in increasing order for 
. Prove that there exists a sequence of distinct prime numbers 
satisfying 
for all 
be real numbers that satisfy the condition:![\[
a_i \cdot a_j \geq i^2 - j^2 \quad \text{for all} \quad 1 \leq i \leq j \leq 2024.
\]](http://latex.artofproblemsolving.com/4/3/9/43970110f50d6fad23211b1ac099c128dca41636.png)
What is the minimum value of 
?
be a cyclic quadrilateral with 
and let 
and 
be the feet of altitudes from 
to the lines 
and 
,.Prove 
bisects 
.
be an acute triangle. Points 
, and 
lie on a line in this order and satisfy 
. Let 
and be the midpoints of 
and 
, respectively. Suppose triangle 
is acute, and let be its orthocentre. Points 
and lie on lines 
and 
, respectively, such that 
and are concyclic and pairwise different, and 
and are concyclic and pairwise different. Prove that 
and are concyclic.
in the plane of a square 
for which ![\[\max (PA, PC)=\frac1{\sqrt2}(PB+PD).\]](http://latex.artofproblemsolving.com/3/9/8/3984b1970837a85af0bac93fd711843630a1fd99.png)
. Assume and are on the same side of 
. Then, 
and Ptolemy on 
finishes. If and 
are on the same side of we can do the same thing. skibidi skibidi
:
