[/quote]
,
and 
be two infinite sequences of integers. Prove that there exists an infinite sequence of integers 
such that for any positive integer 
,![\[
\sum_{k|n} k \cdot z_k^{\frac{n}{k}} = \left( \sum_{k|n} k \cdot x_k^{\frac{n}{k}} \right) \cdot \left( \sum_{k|n} k \cdot y_k^{\frac{n}{k}} \right).
\]](http://latex.artofproblemsolving.com/2/5/9/259ee32c89befe7a3ee2797e0d5b1df1c3066cf6.png)
such that for all reals 
Proposed by Aritra12, India
and 
imply that either 
or 
.
implies that 
.
also implies that 
implies that for any 
,
or 
.
,
is injective. 
and 
imply that 
for all 
.
.
such that 
,
gives 
for all such 
.
then gives 
and thus 
.
which means that 
.
for all 
for some 
,
which is impossible. Thus 
in this case.
let 
be the number whose 
-
-
.
.
and 
.
Prove 
Find maximum 
be reals such that 
and 
Prove that
Let 
Prove that
Prove that
and 
Prove that
Let 
Prove that
,
,
is a 
is 
,
is a 
is also prime
,
,
be the midpoints of sides BC, CA, and AB, respectively. Point 
is located on the ray 
so that 
is similar to 
.
and 
on rays 
and 
,
,
,
are concurrent, i.e. these three lines intersect at a point.
and a constant 
,
and 
are also friendly polynomials. Prove that 
.
Y by jkim0656, DhruvJha, A7456321, Yrock, evt917, Samcharya
richard, thank you for everything, for making this website and creating aops. even though I joined aops relatively recently, there are so many materials and books, and other things that help us all learn, and i think that aops was my most valuable math learning tool. just wanted to say thanks
also just wanted to say that the aops community is probably the most vibrant, widespread, and most enthusiastic people about math ever. like its just amazing
EDIT: guys lets all change our pfps to richards
march 24, 2025 will go down in history as the AoPS Memorial Day, to remember richard and the real ones who made aops
also just wanted to say that the aops community is probably the most vibrant, widespread, and most enthusiastic people about math ever. like its just amazing
EDIT: guys lets all change our pfps to richards
march 24, 2025 will go down in history as the AoPS Memorial Day, to remember richard and the real ones who made aops
This post has been edited 4 times. Last edited by nmlikesmath, Mar 27, 2025, 2:19 AM
