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.
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.
with its center 
. There are two fixed points 
and one moving point 
on . The midpoint of the line segment 
is 
. 
is a fixed point on . Line 
intersects
at 
, and line 
intersects 
at 
. such that is always tangent to 
when varies.
such that:![\[
a^2 + b^2 \mid a^3 + b^3
\]](http://latex.artofproblemsolving.com/a/1/d/a1ddbd20c854e291da5670de449229d1664fa56c.png)
be an isosceles triangle with 
. The circle 
, passing through 
and 
, intersects segment 
at 
. The circle 
is tangent to at and passes through 
. Let and 
be the midpoints of segments and 
, respectively. The line 
intersects and at points 
and 
, respectively, where and are the intersections closer to . Prove that 
.
such that for any 
:
, then either travels [list] units vertically (up or down) and 
units horizontally (left or right); or units horizontally (left or right) and units vertically (up or down)., the ant can choose to go to one of eight possible points.
and 
, the ant can travel from 
to 
using at most 
moves.
, the sides and 
are parallel and the angles 
and 
are acute. Show that it is possible to divide the triangle into 4 disjoint triangle 
and the triangle 
into 4 disjoint triangles 
such that the triangles 
and 
are congruent for all 
.
is given in the plane, let 
be the set of points which lie strictly inside it. Also let 
be the set of circles in the plane. For a point 
, let 
be the reflection of in sides 
respectively. Define a function 
such that 
is the circumcircle of 
. Is injective? is called injective if for any 
, 
board, some squares are painted red. Let be the number of rows that have an odd number of red squares and let be the number of columns that have an odd number of red squares. Find all possible values of 
. For each value found, give a example of how the board can be painted.
sides are marked on the blackboard. Ana and Beto play alternately, Ana begins. Each player, in turn, must do the following:
join two vertices with a segment, without cutting another already marked segment; or delete a vertex that does not belong to any marked segment.
be the set of positive real numbers. Determine all functions 
such that ![\[x \big(f(x) + f(y)\big) \geqslant \big(f(f(x)) + y\big) f(y)\]](http://latex.artofproblemsolving.com/e/7/1/e71b6e0d0b858eb46b81149f1e6be8c41e13d301.png)
for every 
.
with 
=
. You can check the permutation with respectly sastyfing
the permutation satisfying the property is![\[(0,1,n-1,3,n-3,5,n-5....2,n)\]](http://latex.artofproblemsolving.com/3/a/8/3a8464df29cbab99efaa06784100163a776f2687.png)
are 
all distinct
. So it does not hold for even n.
and it cannot be divisible by , so is odd. Now, for any odd consider the construction 
if is odd and 
if is even. It is easy to check that this construction works for all odd .
consists of the numbers 
in some order. For which positive integers 
all have different remainders when divided by 
?
.
do not form a complete residue class modulo 2k+1 since 
and 
are both 
.