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.
, define 
and let 
be the set 
.
we have that 
divisible by 
such that if we define multi-set, 
then 
+1 w
, 
be such points on sides 
, 
of a triangle 
that 
is cyclic. 
, 
is the midpoint of 
. In triangles 
and 
let 
, 
be the 
-humpty points and let , 
be the -dumpty points. Prove that 
and 
.
and 
with 
. Determine the smallest positive integer 
satisfying the following condition:
, 
, 
, there exists a real-coefficient polynomial 
in variables satisfying the following properties: has degree at most ; is not identically 
;
of 
, the equation 
holds.
the inequality holds: 
.
there exists an -digit number divisible by 
all of whose digits are odd.
be non-zero distinct real numbers so that there exist functions 
so that:
and large enough 
.
so that:
and large enough .
be a quadrilateral with both an incircle and a circumcircle. Let 
and 
be the incenter and circumcenter of , respectively. Let 
be the intersection of lines and 
, and let 
be the intersection of lines 
and 
. Let and be the intersections of the line 
with lines and , respectively. Prove that the circumcircle of 
, the circumcircle of 
, and the line 
are concurrent.
such that 
.
vertices is written a real number. Prove that
(with possibly repeating vertices) in which the sequence of numbers is increasing.
be positive integer such that ![\[ \text{LCM}(x+2, y+2)+\text{LCM}(x, y)=2\text{LCM}(x+1, y+1).\]](http://latex.artofproblemsolving.com/7/3/3/7338e4d6058ededa9b8cd9fcc945ae362db4bad4.png)
Prove that is divisible by .
be the incircle of an arbitrary triangle with tangency points 
on 
and respectively. intersects at 
. Segments 
and 
meet respectively at and 
. Show that 
and 
are concurrent.
and 
are harmonic, so by harmonic ratios if we let 
and we want to prove 
collinear , we have to prove 
we do the same thing for the other ratio , so both are equal to 
and were done
and are harmonic, so by harmonic ratios if we let and we want to prove collinear , we have to prove we do the same thing for the other ratio , so both are equal to and were done
such that
for any 
,
