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 relatively prime integers 
and 
such that 
is divisible by 
.
![\[\frac{a}{b^3+4}+\frac{b}{c^3+4}+\frac{c}{d^3+4}+\frac{d}{a^3+4}\]](http://latex.artofproblemsolving.com/4/d/2/4d21982ba412c0bb4a7985798e81bc5b01ac0e43.png)
given that 
, 
, 
, 
are nonnegative real numbers such that 
.
be the set of all integers. Find all pairs of integers 
for which there exist functions 
and 
satisfying![\[ f(g(x))=x+a \quad\text{and}\quad g(f(x))=x+b \]](http://latex.artofproblemsolving.com/3/2/8/328146c3331a66cdc5cfc4a01b0609e91d278ec5.png)
for all integers 
.
satisfying 
.Prove that:
such that 
be the sequence defined by 
and 
for each integer 
. Suppose that 
is a prime and 
is a positive integer. Prove that some term of the sequence is divisible by 
.
is interesting if 2018 divides 
(the number of positive divisors of ). Determine all positive integers such that there exists an infinite arithmetic progression with common difference whose terms are all interesting.
be integers and 
be a prime for which![\[ x^2-3xy+p^2y^2=12p \]](http://latex.artofproblemsolving.com/2/4/c/24cb51eb55144a9a1e0e489d115993496a052e79.png)
.
satisfying 
for all positive integers .
be an isosceles trapezoid with 
. Let 
be the midpoint of 
. Denote by 
and 
the circumcircles of the triangles 
and 
, respectively. Let 
be the crossing point of the tangent to at 
with the tangent to at 
. Prove that 
is tangent to .
, let 
denote the feet of the altitudes from 
, respectively, and let be the circumcircle of 
. Let 
and 
be the circles through tangent to at and 
, respectively. Show that and meet at a point on 
other than .
such that![\[ f(x+f(y))=y+f(x+1),\]](http://latex.artofproblemsolving.com/b/a/3/ba35828951778c0d35f50b5b557796bd2cf9e338.png)
for all 
.
is defined by 
, and 
for 
. Consider the set 
of ordered pairs 
for which there is a finite set 
of positive integers such that 
, 
. Prove that there exist real numbers 
, 
, and 
with the following property: An ordered pair of nonnegative integers satisfies the inequality ![\[m < \alpha x+\beta y < M\]](http://latex.artofproblemsolving.com/a/9/b/a9bb35658173fdab821cc8180420e47d5437e0a2.png)
if and only if 
.
.
, satisfying ![\[
f(xy)(f(x) - f(y)) = (x-y)f(x)f(y)
\]](http://latex.artofproblemsolving.com/0/9/b/09b8ed4e133ccf33b3f902fd9a9aa8e23c264341.png)
for all 
.
