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.
such that for all 
,
such for any 
:
.
be an acute triangle with incentre 
and 
. Let lines 
and 
intersect the circumcircle of at 
and 
, respectively. Consider points 
and 
such that 
and 
are parallelograms (with 
, and 
). Let 
be the point of intersection of lines 
and 
. Prove that points 
, and are concyclic.
such that 
. Prove that 
be a positive integer. On the table, we have 
ornaments in different colours, not necessarily of each colour. Prove that we can hang the ornaments on Christmas trees in such a way that there are exactly ornaments on each tree and the ornaments on every tree are of at most 
different colours.
and point 
, if the cevian triangle and the pedal triangles of to are orthologic, then prove that the cevian triangle and the reflection triangle of to are also orthologic.
and 
are orthologic if the perpendiculars from the vertices 
, 
, 
on the sides 
, 
, and 
are concurrent. Furthermore, if this is the case, then the perpendiculars from the vertices 
, 
, 
on the sides 
, 
, and 
are also concurrent. over 
, 
and 
.
+2 w
there are elements 
in an arithmetic progression in the order 
such that 
.
such that the value of the
is a square of an integer.
be reals such that 
. Prove that
Let be reals such that 
. Prove that
consecutive natural numbers starting from 
on whiteboard. Every minutes, he will erase 
numbers (not necessary different), in the form of 
and write 
numbers back on the whiteboard in the form of 
. Tony will stop whenever he cannot erase any 4 numbers that satisfy the condition anymore. If let's say, Tony starts at 
(8 AM), prove that Tony will stop before 
(4.30 PM)
Prove that
. Prove that
and 
.
(and prove):
.
meets the side 
at point 
such that 
.
.
on 
is similar to 