[/quote]
,
denote the set of all real numbers. Find all functions 
such that
for all real numbers 
,
.
,
,
of an acute-angled triangle 
intersect at point 
.
and 
of triangle 
meet the segments 
and 
at points 
and 
respectively. Denote the value of the angle 
by 
.
and 
similarly. Find the sum 
.
.
be any other point on the arc 
of the circumcircle of 
intersect line 
at 
and let line 
intersect line 
at 
.
and 
)
intersect the circumcircle of 
again at 
)
concur.
at 
,
meets the incircle again at a point 
.
and 
meet the incircle of triangle 
and 
,
and 
are concurrent.
be an integer. We call a pair of integers 
g![\[0\leqslant a<k,\hspace{0.2cm} 0<b \hspace{1cm} \text{and} \hspace{1cm} (a+b)^2=ka+b\]](http://latex.artofproblemsolving.com/e/8/d/e8df562e024b8476d69fdaea15338c27bb6c4eb8.png)
Prove that the number of 
.
written on the first page and 
empty pages. Suppose in each of the next 
be the largest possible number such that for any 
,
on the last page of her grimoire. Prove that there exists a positive integer 
,![\[n^{0.99n}\leqslant F(n)\leqslant n^{1.01n}.\]](http://latex.artofproblemsolving.com/5/7/e/57e2b7e3d97564665d8de28a3afd74bdcf74ba5b.png)
.
intersect the incircle of triangle 
.
is perpendicular to line 
.
lies on segment 
is tangent to line 
.
such that 
.
.
,
be its circumcentre, and let 
and 
and 
respectively. The circumcircles of triangles 
and 
.
and 
then 
is tangent to the circle 
,
,
,
be distinct positive integers, 
.
and 
such that 
does not divide any of the numbers 
,
,
.
(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"
