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.
and 
be positive integers with 
such that 
divides 
. Prove that
and 
; or
and 
.
with their incenter 
. Point 
lies on 
on the same side as 
wrt 
. Point 
lies on the shorter arc 
of the circumcircle . It is given that 
Prove that 
is the angle bisector of 
.
. We call a distribution the act of distributing the integers from 
represented by tokens to players 
to 
so that they all have the same number of tokens in their urns.
beats 
when, when and each draw a token from their urn, has a strictly greater chance of drawing a larger number than . We then denote 
. A distribution is said to be chicken-fox-viper when 
What is 
such that the following equations are both true:
be reals such that 
Prove that
of the triangle touch its side at 
. Let the line 
intersect at 
and denote the excentre of opposite to 
by 
. Let 
and 
be the midpoints of and 
respectively.
and 
are concyclic.
be a triangle with 
. is the midpoint of 
. Let the angle bisector of 
cut 
at 
and 
be the centroid of . 
meets at 
and 
is the projection of 
onto . Prove that 
lie on a common circle.
be 
. Points 
and 
are two moving points on the circle 
: 
, satisfying 
. Find the maximum area of 
.
. What is the least possible positive integer value of 
such that there exists a non-negative integer 
for which the set 
is fragrant?
is a friend of , then is a friend of .) designates as her best friend, then (unfortunately) it does not follow that necessarily designates as her best friend. Someone designated as a best friend is called a -best friend. More generally, if 
is a positive integer, then a user is an 
-best friend provided that they have been designated the best friend of someone who is an 
-best friend. Someone who is a -best friend for every positive integer is called popular.
such that the equation![\[f(xf(x)+y) = f(y) + x^2\]](http://latex.artofproblemsolving.com/2/0/a/20a245460634f0455fc703d7ce41c475c1ae4970.png)
holds for all rational numbers 
and 
.
denotes the set of rational numbers. a grid of 9*9, we choose a little square in of this grid three times, we can choose three times the same.
If you just only draw on your paper (and maybe wrong) and think that wrong, you should prove your statement before saying something. If you draw on computer, please give me the figure.
to 
same with 
, it's will be IMO 2009 problem 2. so, any idea now, I still thinking and cannot find how to apply that problem for this 
be midpoints of 
，
，
，
be midpoints of 
，
be the circumcircle of 
，
be parallelogram，
，
.
，
，the 9-point center 
is the circumcircle of 
，so， is midpoint of 
.
. On another hand，we can easy found 
is a isosceles trapezoid，and 
， is on perpendicular bisector of 
，the same way， is also on perpendicular bisector of 
. is midpoint of 
，therefore，we get 
. Q.E.D
be midpoints of ，，， be midpoints of ， be the circumcircle of ， be parallelogram，， .，，the 9-point center is the circumcircle of ，so， is midpoint of . . On another hand，we can easy found is a isosceles trapezoid，and ， is on perpendicular bisector of ，the same way， is also on perpendicular bisector of . is midpoint of ，therefore，we get . Q.E.D
,
,
,,
is the food of the perpendicular line from to .
is the line pass throught and paralley 
, meet 
at 
and meet at , we have is the midpoint of 
,
meet at , apply "Butterfly Theorem" we have is the midpoint of 
so we have 
so 
![\[ Q.E.D\]](http://latex.artofproblemsolving.com/9/9/4/9944bb829d83b6c8c3731e0b66fafd134f718701.png)
is the midpoint of 
is the mid point of , you should read "Butterfly Theorem" again. is the midpoint of because 
and is midpoint of 
be a triangle and its circumcircle. 
. Let be the midpoint of , be the midpoint of 
, be the midpoint of . the circumcircle pass through 
meets at . Prove: 
and 
Show that
Where 
Where 
