[/quote]
,
be non-negative real numbers such that 
Prove that
be circumscribed about circle 
,
be the orthocenter of 
.
at 
.
at 
.
be the midpoint of 
,
meet 
.
at 
.
is tangent to 
such that![\[ f(x)f\left(x + 4f(y)\right) = xf\left(x + 3y\right) + f(x)f(y) \]](http://latex.artofproblemsolving.com/d/b/1/db14b64aa71cb3c8dbf53707f719f5d9a43bd7e4.png)
for any positive real numbers 
.
and 
be coprime integers. Call a set 
balanced if the numbers 
form an arithmetic progression with difference 
,
form an arithmetic progression with difference 
.
and 
,
be an integer. The sequence 
is defined via 
,
and![\[ a_{n+2} = ka_{n+1} + a_n \]](http://latex.artofproblemsolving.com/b/5/e/b5e3e6f9f0b0709113498266e41b372299a2986f.png)
for any positive integer 
of positive integers such that 
.
be a circumscribed quadrilateral with incircle 
and no two opposite angles equal. Let 
be an arbitrary point on the diagonal 
,
and 
intersect 
and 
.
has exactly 
ordered pairs 
as solutions?
.
be positive reals such that 
.
(Shining Sun, USA)
be an acute triangle with 
,
of side 
)
.
and 
,
,
.
intersects line 
at a second point 
.
and 
are perpendicular.
such that exactly 
of the integers 
are squarefree.
be a non-constant monic polynomial with integer coefficients and let 
be an infinite sequence. Prove that there are infinitely many primes, each of which divides at least one term of the sequence 
.
cells of an 
square grid are colored black, and the remaining cells are white. The cost of such a coloring is the minimum number of white cells that need to be recolored black so that from any black cell 
,
through a sequence 
of black cells where each consecutive pair 
are adjacent (sharing a common side) for every 
.
denote the maximum possible cost over all initial colorings with exactly 
such that![\[
\frac{1}{3}n^{\alpha} \leq f(n) \leq 3n^{\alpha}
\]](http://latex.artofproblemsolving.com/0/a/0/0a06d4ff52479c8cb785c6450cea8748bf7769b6.png)
for any 
.
Y by PikaPika999
For each positive integer 
, define
and
meaning 
is the least common multiple of the numbers 
. Determine whether or not there exists a constant 
such that
for every positive integer .
, define
and
meaning 
is the least common multiple of the numbers 
. Determine whether or not there exists a constant 
such that
for every positive integer .This post has been edited 2 times. Last edited by EmersonSoriano, Apr 5, 2025, 7:43 PM
Reason: change
Reason: change
and that for 
we have ![$(n +k,n+l)=(n+k,n+l,k-l)|[1,\cdots,2015]=:M$](http://latex.artofproblemsolving.com/8/e/2/8e26059b4a9def6738db48be48bfeb46ed1fbad5.png)
.
is prime, then 
for at most one 
.
be the set of prime numbers 
then 
.
with 
,
then 
when 
then 
.
then 
.
for some 
