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.
interesting if 
is composite and for every divisor 
of at least one of 
and 
is also a divisor of with 
such that 
holds for all real numbers 
and 
.
such that the numbers ![\[ (a_1)^{2018}+a_2, (a_2)^{2018}+a_3, \dots, (a_{2018})^{2018}+a_1 \]](http://latex.artofproblemsolving.com/6/1/a/61ad92b49bd9b4b8839e6f6c4e81c758a11835a3.png)
are all powers of 
be a triangle with 
. Let 
be the midpoint of 
, and let 
be a point on segment such that 
. Let 
be the point of intersection, different from 
, of the circumcircle of triangle 
and line 
. Define 
and 
as the points of intersection of line 
with 
and 
, respectively. Prove that is the midpoint of 
.
have orthocenter 
and altitude 
with 
on side 
. Let 
be the midpoint of side , and let 
be the reflection of over . Let 
be a point on line 
such that lines 
and are parallel, and let the circumcircles of 
and 
meet again at 
. Prove that 
.
be a set of integers satisfying the following property: For every positive integer 
and every set of coefficients 
, all integer roots of the polynomial 
are also elements of . It is given that contains all numbers of the form 
where 
are positive integers. Prove that contains all integers. students and their test results on 
different subjects, we say that student 
dominates student 
if and only if outperforms on all subjects. Assume that no two of them have the same score on the same subject, find the probability that there exists a pair of domination in class.
of positive integers that satisfy the equation![\[
x + xy + xyz = 31.
\]](http://latex.artofproblemsolving.com/a/7/8/a78d1e9d66831f11e7de08c109855e7727ec8535.png)
![\[\left(-\cos(1^{\circ})\right)\left(-\cos(2^{\circ})\right)\left(-\cos(4^{\circ})\right) \cdots \left(-\cos{(256^{\circ})}\right)\]](http://latex.artofproblemsolving.com/4/4/e/44e5bbdf821aed7bb2d855efcf39af763ebaea48.png)
on a blackboard. Then, he erases each term on the blackboard with probability 
. What is the expected value of the remaining expression?
that satisfy the following condition: For every integer k such that 
there always exists a positive integer that is divisible by n and 
. 
: sum of elements of 
. Also please mind your notation: it should be 
and in the last line.
. Choose 
prime and 
. Then 
and this runs through all residue classes mod .
of exists with 
. Then we finish the problem as before.
, the prime factorisation.
and write 
for the set of indices 
with 
. We write 
for the rest, hence the set of indices 
with 
. These sets may be empty.
such that 
for 
and such that 
for 
. This is not difficult: write 
and take 
. In particular we can choose 
and we set 
.
, a multiple of .
and one checks that .
. Also please mind your notation: it should be and in the last line.. Choose prime and . Then and this runs through all residue classes mod . of exists with . Then we finish the problem as before., the prime factorisation. and write for the set of indices with . We write for the rest, hence the set of indices with . These sets may be empty. such that for and such that for . This is not difficult: write and take . In particular we can choose and we set ., a multiple of . and one checks that ., is this right? When I try with some case it is wrong.
that satisfy the following condition: For every integer k such that there always exists a positive integer that is divisible by n and . : sum of elements of 
,
and 
be integer numbers. Prove that:
is divisible by 
