[/quote]
,
such that:
for any positive integer 
,
are all distinct, and
,
?
.
be its incenter, 
be its 
-
be midpoint of arc 
,
be the intersection of the line 
with 
,
be the intersection of the angle bisector of 
with the line 
,the intersection of line 
with 
and 
and 
such that 
.
are collinear. Who has a winning strategy?
and let 
be positive integers such that 
.
and determine when equality holds.
different ropes. She then performs a experiment that causes 
children, each connected to each other with the same type of rope, to be cured. Two experiments are said to be of the same type, if each of the ropes connecting the children has the same medicine imbued in it. She then unties them and lets them go back home.
be the minimum m such that Myla can perform at least 
experiments of the same type. Prove that:
For every 
there exists a 
and 
such that for all 
,![\[f(n, k) = a_kn + b_k.\]](http://latex.artofproblemsolving.com/6/6/0/660699c6cec811eee25f727076665a6df3257f57.png)
Find the value of 
for every 
.
intersect 
at 
.
be the Lemoine points of triangle 
respectively. Prove that 
are collinear.
be a sequence of positive real numbers such that for every 
,![\[
a_n = \max_{1 \leq i \leq 2025} a_{n-i} - \min_{1 \leq i \leq 2025} a_{n-i}
\]](http://latex.artofproblemsolving.com/e/3/0/e30e9e533eb9a41cf847484e676ac4522edda665.png)
Prove that there exists a natural number 
such that for all 
,![\[
a_n < \frac{1}{1404}
\]](http://latex.artofproblemsolving.com/0/1/f/01f1a81eed2230332d51a15501ef2a6ad3a7ec82.png)
with no obtuse angles.![$$\frac1{\sqrt3}\cdot\left(\cot A+\cot B+\cot C\right)+\left(2-\sqrt 3\right)\sqrt[3]{\cot A\cot B\cot C}\geq\frac2{\sqrt3}.$$](http://latex.artofproblemsolving.com/d/1/4/d14154281d7ccf60a2dcfe1d937b974ae1d74f97.png)
and 
lie on 
.
and 
such that the circumcenter of triangle 
coincides with the circumcenter of triangle 
.
such that : 
such that : 
and 
such that 
.
written on the first page and 
be the largest possible number such that for any 
,
,![\[n^{0.99n}\leqslant F(n)\leqslant n^{1.01n}.\]](http://latex.artofproblemsolving.com/5/7/e/57e2b7e3d97564665d8de28a3afd74bdcf74ba5b.png)
of positive integers for which there exist positive integers 
and 
holds for all integers 
(Note that 
denotes the greatest common divisor of integers 
and 
)
.
Y by
Let 
be a triangle and the points 
and 
on 
, 
and 
on 
, and 
and 
on 
be such that 
and 
. The intersections of 
with 
and 
are 
and 
respectively, and the intersections of 
with 
and 
are 
and 
, respectively. Prove that the lines 
and 
are concurrent.
be a triangle and the points 
and 
on 
, 
and 
on 
, and 
and 
on 
be such that 
and 
. The intersections of 
with 
and 
are 
and 
respectively, and the intersections of 
with 
and 
are 
and 
, respectively. Prove that the lines 
and 
are concurrent.
,
,
,
.
,
,
multiplying everything gives 
So from Menelaus 
are collinear. Since 
,
,
are collinear 
are concurrent(From Desaurges Theorem). So 
are collinear. Similar for 
and 
.
, 
,
are collinear 
are concurrent(From Desaurges Theorem). We are done.
and let 
.
passing through the point 
,
,
,
.![[asy]
import geometry;
size(10cm);
pair A=dir(120);
pair B=dir(210);
pair C=dir(330);
pair L= .2A+.8B;
pair K=.8A+.2B;
pair N=.3B+.7C;
pair M=.7B+.3C;
pair P=.25A+.75C;
pair Q=.25C+.75A;
pair R=intersectionpoint(line(K,N),line(M,Q));
pair T=intersectionpoint(line(K,N),line(L,P));
pair D=intersectionpoint(line(P,N),line(M,L));
pair E=intersectionpoint(line(P,N),line(K,Q));
pair X=intersectionpoint(line(E,B),line(D,R));
pair F=intersectionpoint(line(K,Q),line(L,M));
pair S=intersectionpoint(line(M,Q),line(L,P));
draw(conic(K,L,M,P,Q),deepgreen);
draw(B--E,grey); draw(A--D,grey); draw(C--F,grey); draw(N--K); draw(Q--M); draw(L--P);
draw(A--B--C--cycle, red);
draw(D--E--F--cycle);
dot("A",A,dir(A));
dot("B",B,dir(B));
dot("C",C,dir(C));
dot("K",K,dir(120));
dot("L",L,dir(185));
dot("M",M,dir(250));
dot("N",N,dir(-50));
dot("P",P,dir(0));
dot("Q",Q,dir(30));
dot("R",R,dir(10));
dot("T",T,dir(60));
dot("D",D,dir(280));
dot("E",E,dir(20));
dot(X);
dot("F",F,dir(165));
dot("S",S,dir(120));
[/asy]](http://latex.artofproblemsolving.com/b/1/4/b148db34973f8fc27cf21c4755da3ff5bd2cf2dd.png)
