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.
+2 w
be a cyclic quadrilateral in which the lines 
and 
meet at a point 
. Let 
be the point of the line 
, different from 
, such that 
. We construct the parallelograms 
and 
. Prove that the points 
lie on the same circle.
defined on the set of all positive integers and taking non-negative integer values, satisfying the three conditions:
for at least one 
;
for every positive integers 
and 
;
there are infinitely many positive integers such that 
for all 
.
be positive real numbers such that 
. Find the minimum value of the following sum :
knowing that the denominators are positive real numbers.
point 
was chosen such that 
and 
. Prove that if perpendicular from to passes from the intersection of diagonals of , then 
.
balls to 
boxes as she wishes. After that, Aslı and Zehra make alternated moves which consists of taking a ball in any wanted box starting with Aslı. One who takes the last ball from any box takes that box to herself. What is the maximum number of boxes can Aslı guarantee to take herself regardless of Zehra's moves?
, 
. The internal angle bisector of 
and the external angle bisector of intersect the ray at points 
and , respectively. Given that 
, prove that 
.
![\[f(xf(y) + x) = xy + f(x)\]](http://latex.artofproblemsolving.com/9/9/f/99f580ebc50846e6bc2c004667559922749a4dfa.png)
and 
intersect at two points 
and 
. Let 
be the line tangent to these circles at 
and 
, respectively, so that lies closer to than . Let 
be the line parallel to and passing through the point , with 
on and 
on . Lines 
and 
meet at 
; lines 
and meet at 
; lines 
and meet at 
. Show that 
.
be an acute triangle. Let be the altitude on , and let 
be any interior point on . Lines 
, when extended, intersect 
at 
respectively. Prove that 
.
has centre 
and touches the sides 
, 
, 
at points 
, 
, 
, respectively. Let 
, 
, 
be excentres of triangle , touching the sides , , respectively. The segments 
and 
intersect at 
. Similarly, segments 
and 
intersect at 
, and the segments 
and 
at 
. Prove that 
is the center of the circumcircle of the triangle 
.
positive integers with mean, median, mode, and range equal to positive real 
Compute the minimum possible value of 
