[/quote]
,
be a cyclic quadrilateral in which the lines 
and 
meet at a point 
.
be the point of the line 
,
,
.
and 
.
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 
for all 
.
be positive real numbers such that 
.
knowing that the denominators are positive real numbers.
was chosen such that 
and 
.
.
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?
,
.
and the external angle bisector of 
and 
,
.
![\[f(xf(y) + x) = xy + f(x)\]](http://latex.artofproblemsolving.com/9/9/f/99f580ebc50846e6bc2c004667559922749a4dfa.png)
and 
intersect at two points 
and 
.
be the line tangent to these circles at 
and 
,
be the line parallel to 
on 
on 
and 
meet at 
;
and 
;
and 
.
.
be an acute triangle. Let 
be any interior point on 
,
at 
respectively. Prove that 
.
has centre 
and touches the sides 
,
,
at points 
,
,
,
,
,
be excentres of triangle 
and 
intersect at 
.
and 
intersect at 
,
and 
at 
.
is the center of the circumcircle of the triangle 
.
(i dont remember exactly if it was 8032038 or some other number, so correct me on this)
(i posted this at 12:29) lol
?
Y by RainbowJessa, cubres, At777
I believe discussion is allowed after yesterday at midnight, correct?
If so, I will put tentative answers on this thread.
By the way, does anyone know the answer to Geometry Problem 5? I was wondering if I got that one right
Also, if you put answers, please put it in a hide tag
If so, I will put tentative answers on this thread.
By the way, does anyone know the answer to Geometry Problem 5? I was wondering if I got that one right
Also, if you put answers, please put it in a hide tag
This post has been edited 21 times. Last edited by Aaronjudgeisgoat, Apr 22, 2025, 11:32 AM
minutes to solve it.
game using the integers 
exactly once"
